Since you have a RAM 1500 I'll assume you have P-series tires. Tires are not usually the weak link compared to the rear axle until you get to a 2500 truck. Either way, the pressure printed on the driver's door is the pressure that will give you the weight carrying capacity Dodge has specified for that truck, probably 35 psi?? P-series tires can generally be inflated to 44 psi, but that doesn't increase their load capacity. That will give you a harsher ride, but possibly better handling as well as a P-series tire is quite soft and often on a less than optimal wheel width. However, too much pressure will decrease the tire patch on the road and also wear the centers faster especially when your truck is lightly loaded.
I had LT225/R16 Load C tires on my previous 1500 and ran them at 50 psi front 60 psi rear most of the time. This tire is a little shorter than the P235 which throws off the speedo a bit, but its performance and handling is a huge improvement over the factory P235 albeit with a harsher ride.
If you have LT tires you can match their pressure to the load you'll be carrying which is how they are designed. Air them down for solo driving, air them up for towing. You cannot do this with a P tire. You still can't air them down too much or you'll get soft mushy handling, lower gas mileage and wear the outside edges. In any case you must build in a generous allowance and never risk under inflation.
Here is a link to Goodyear's Load Inflation Table:
Google Load Inflation Table to find links to most other manufacturers' tables, but tire sizes/pressures/loads are standard.
Tirerack.com is also a wealth of information.
Strictly speaking you can play with pressures the same way with ST tires, but in practice the trailer manufacturers' have left less room. The ST205/75R15 on my 2499 have a load capacity of 1820 lb at 50 psi and 1720 lb at 45 psi. Anything less and I'm already under the 7000 lb GVW the 4 tires have to carry. However, those tires don't carry the tongue weight, but I just don't feel good about messing with them and will never run them at less than 45 psi.