We have a small stall shower in an apartment building. If you look down the drain you see water like 3 inches down. Our plumber came, ran the water, it did not overflow into the stall and he said it was normal that what we are seeing is the trap. Is this correct? Will this never be a problem?

Plumber is correct and what you see is not only normal, but is required. A trap is a low place in a drain pipe for water to settle. The water that settles in a trap acts as a seal to prevent sewer gasses from escaping. All plumbing fixtures are required to have a trap. Look under your kitchen sink.

Yep, very surprised the plumber did not bother to point this out. There is always water in a trap if it's working properly