A couple of months ago, one of the dealers near where I live showed me a 1878 morgan that he picked out of a batch of average circulated common date morgans. While he was getting the coin out from the safe, he told me that he believed that it may be a proof. When he handed the coin over to me, I knew that I wasn't any ordinary 1878 morgan. It had a needle sharp strike, sharp squared off rims, and reflective surfaces. Unfortunately, it had been cleaned as if it had been aggressively wiped. I told him what I thought and apparently another dealer agreed since he said that he got a grand for it.
Like SsuperDdave said, there is no fool proof method. Much of it comes from experience. I don't consider myself a morgan expert but there are a few things that I look for. One thing I do is to check the mintmark. I know the presence of a mintmark does not automatically rule out a proof but branch mint proofs are very rare and it is much more likely its a PL/DMPL business strike than a proof. Like I mentioned above, I also check for a sharp strike with squared off rims and reflective fields. Watch out if the coin is reflective but there are halos around the devices and the luster/surface within those halos looks decidedly more business strike than proof. While most proofs have sharp details, not all do. For example, the 1893 proof morgan sometimes looks like it was struck in New Orleans. Like in many areas of numismatics, there are always exceptions to the rules and as already mentioned even experts can't always agree. Take for example the 1841 quarter eagle.