Is the word problem for regular languages in ALogTime?
If the regular expression is not fixed, it is rather doubtful that your problem belongs to ALOGTIME. However, the phrase word problem usually refers to a fixed setting, which in this case means that the regular language is fixed.

If the regular language is fixed, this is apparently shown in Barrington's famous paper. Alternatively, you can describe a logtime game for deciding the word problem for regular languages. The game starts with the YES player stating the final (accepting) state and the state at the middle (the initial state is of course fixed). The NO player then chooses one of the halves, and the game continues until it comes down to a word of length 1, in which case the problem can be settled by examining one character of the input.