There's absolutely no scientific evidence to suggest wearing any type of hat affects your hair loss directly. This is not to say you should wear anything for too long. If your hair is already thinning, it's more likely to fall out of the roots. At this stage, hair is more likely to be pulled out by combing or brushing. Similarly, if you are regularly taking a hat off and then putting it back on, pick a style which is not likely to catch on thinner hair and pull it out. If there's a band or other part of the hat that could catch, keep the hat on for longer, removing it only when you're sure you will not be wearing it again.
Although everyone claiming expertise will tell you genes are genes and, if they start thinning your hair, that's got nothing to do with your hat. But if you chose to wear a hat which is a very tight fit - perhaps fearing it might blow off in a strong wind - this could reduce the circulation of blood through your scalp and damage the follicles. You would always know when this was a risk because wearing the hat would be quite painful. Similarly, if you were to wear a hat made out of a fabric that did not encourage good ventilation, you might sweat more. Experts suspect that excessive sweating may slow down the growth of hair. But this is slightly speculative.
As to appearance, wearing a hat for too long, can make the hair lie flatter against the head. The longer the hair, the more pronounced the flattening and the result can be your hair looking thinner. Experience also suggests you're more likely to find split ends and other damage. In all this, don't forget the problem really lies with your genes and not the hats. Propecia is the ultimate answer no matter what your decision on covering up the problem. So long as you continue to take the Propecia, the hair loss will stop and, in some cases, will regrow. But when you stop, have the hat ready because the hair loss will resume.