BenB has pointed you to some good info. I'll add some more. You asked a very simple question, but the answer while doable is not an easy one.
The camper roof can one area of water intrusion. And in many cases, the worst area. But it is not the only area.
I will talk about the roof first. You said,
Fortunately it has never had a leak. Looking at the seams, although, it doesn't look like any updated caulking has taken place, although we haven't climbed on the roof yet to check it out, the side seams look original.
I'm not trying to be a wise guy here, just trying to help and educate you so you can keep your camper dry and as long as you want to. How do you know it has not had a leak or does not have one now? Water leaks have tricked many a good camper before. They cannot see inside any signs of water leaks, so they think they never had a leak. Leaks inside the ceiling, walls and floors may exist, just you do not see them, yet. I'm not saying you have a wet camper, just it will take some inspection work to know for sure your camper has no current leaks or wet spots. It can often take 1 to 2 years before signs of a leak show up inside the camper pending what kind of leak you have. A slow one or a larger one. Small ones may take many years to show up inside.
When you said "Looking at the seams, although, it doesn't look like any updated caulking has taken place, although we haven't climbed on the roof yet to check it out, the side seams look original." If on the roof, you do not see any seam caulking touch up's on a 2003 camper, or brand new caulk, unless the camper was stored inside all it's life, odds exist your camper may not be 100% dry.
Many camper folks do not know the RV roof has special maintenance needs. As such they do not know what needs to be done routinely to maintain an RV rubber roof. RV roofs need to have routine maintenance done on the roof systems to wash and clean them along with caulking checks and touching up tiny splits in the caulking when caulking cracks start. The Dicor recommendation (the company who makes the roofing material) for roof cleaning is 4 times a year. And I'll add when cleaning, that same 4 times a year is when the caulking needs to be checked and touched up. If the camper is stored outside, going into winter and in the spring coming out of winter roof check is critical as winter can be one of the biggest times of year when cracks start in the caulking. The other 2 times spread across the hotter months of the year is good practice.
I will also add, using a UV protectent on the clean rubber and the caulking will help extend it's life. Dirco now states it can be used they even now sell a UV product. I have been using 303 Aerospace UV protectent since 2004 and I can attest it helps. This stuff https://www.goldeagle.com/product/30...ace-protectant
They sell it in many places. Auto parts stores, Walmart, Amazon, Marine stores etc.
I bought my first Sunline T2499 new in 2004. After year 2.5 tiny splits the size of a hair line split, started showing up. I touched them up as soon as they started as they are not a problem then. We sold that camper and in 2007 bought the camper we have now, the T310SR. It is also a 2004 model. Since it was that new, large damage to the roof was not there yet but the prior owner did not give the roof the TLC it needed. So I started it. That roof today is still in perfect shape. The T310SR had to live outside since is was born in Nov 2003 until Sept 2013 when the new pole barn came. In 2010 I did the Eternabond upgrade to all roof seams as I could not control the caulking cracks across the winter outside. The roof is still leak free and we intend to keep the camper a good long time yet.
If you are considering the Eternabond upgrade, I suggest before doing this you get a moisture meter and confirm you have no leaks in the ceiling or side walls as if a repair is needed, the E bond will have to come off to do the repair. See here for more info on the meter. http://www.sunlineclub.com/forums/f7...per-17613.html
You will need to scan the entire ceiling and walls up at the ceiling along the edges inside to scan for moisture.
The next area is the walls and the putty tape sealant used to seal windows, doors, cargo storage doors etc. RV manufactures use putty tape on the corners, windows, doors etc anything not next to the rubber membrane of the roof. On the rubber roof they use butyl sealing tape. The problem with putty tape is, over time it dries out, splits, separates and creates a leak path to the inside of the camper. This area is the next largest water intrusion location. Basically any penetration into the siding of the camper, is a leak potential sealed with putty tape. While having a putty tape leak point does not mean it is wet inside the camper in that location, but it can be if the right conditions exist. I'll list a few examples.
The front window in your T1950 is a highly probably leak point. When the putty tape dries out and cracks, a leak path is created. When towing in the rain, that water beats in on the front window seal and water can and has entered the camper walls. And since the front wall is also one area of the roof water run off, if the camper is stored nose down, then water runs off the roof down onto the front window before it falls to the ground. A split or separation large enough in the putty tape then can allow water to get it. If you have the moisture meter, you can scan the wall all around the front window and see if the inside wall has moisture in it.
The corner moldings are another source of water entry and a often found one. Same thing happens here with the putty tape, it cracks, splits and then waits for a water to beat against that split until water seeps in. This can be from roof run off, to a gutter blasting water down the side of the camper or even towing in the rain.
Same goes for cargo doors, any other window, entry door etc. Some are worse then others pending the location on the camper and where water runs off the roof. The moisture meter can be used to scan all the walls and floor by going under the camper and placing it on the black waterproof membrane.
These potential leak areas can be addressed on an older camper. The windows, doors, moldings etc can be removed and new fresh butyl sealing tape installed. It costs a little more then putty tape but is a better sealing tape and lasts longer over time. And on the next year from installing, then a bead of Dicor non leveling caulk can be added over the exposed butyl caulk to seal out dirt and add a second layer of water protection.
This link will show how I do the windows. This same process goes for doors, corner moldings etc. This post is on a frame repair, but it also shows the putty tape replacement.
This link will show how the putty tape fails over time. The dirt marks show where the putty tape has let go and a void to the inside has started.
This post shows one method on how to apply the non leveling Dicro caulk on the exposed butyl tape on the next year along moldings, doors, windows etc. to create the 2nd layer of protection against water intrusion.
I am on a mission to redo all the putty tape on every door, window, corner molding etc. replace it with butyl sealing tape and then Dicor caulk as a 2nd layer of protection. I am about 60% of the camper done at this point. This is something that takes time and may take a year or more to complete. You do the worst areas first then the rest later. If the camper is 10 years old, the putty tape has already started to crack, split and separate from the siding and molding. My camper is 13 years old and I can see the cracks on the putty tape all over it. By staying on top of it, I can replace it before a leak starts.
Storing the camper inside while not camping is the best if by good fortune you have that option. Next best is winter storage to cover the camper or find a place to rent inside winter storage. And last is having the camper live outside exposed to snow and ice. Even hard winter rains in the southern states are a problem. In this case making sure all roof caulking is in 100% OK condition going into winter is your best bet. And then be right on top of it come spring in case something showed up over the winter.
If you are interested, they do have a leak checking procedure called a Sealtech test. They pressurise the camper with air pressure and then spray a water & soap solution all over the camper to find leak spots. This way at the time of the test, you can at least find all the potential leak locations that exist at the time of the test. This post, about 1/2 way down this link, will show info on the Sealtech test http://www.sunlineclub.com/forums/f7...tml#post136675
This caulking and sealing may be a lot more then you first thought, but it does provide one solution to what you asked which is how to keep your camper dry and not get a leak. Becoming educated on how campers leak, is also something that can help you know where to look and then prevent problems before they start.
We are here to help. Think over what I said above and ask more questions to better understand the issues at hand. You may not be able to do all the upgrades I suggested right away, but maybe can work towards them in time.
Hope this helps