We are going to finish our basement and our electrical service panel in the corner of a room where we would like to build a new bathroom.
Can we have the service panel in a closet in the new bathroom or would that be a hazard?

A panelboard shall not be located in a bathroom, as per the National Electric Code 2002 Edition Section 240.24(E). But you do mention in a closet, again a panelboard is not allowed in "clothes closet", per NEC 2002 Sec 240.24(D).

If you were able to build a small room, with a door from the bathroom into this new small room, and insure that the panel will have a min of 30 inch wide dedicated space, and 36 in of clear space in front of the panel, and able to close the door leaving the bathroom behind you, should be able to pass inspection.

Before you begin building, you should create a neat sketch, with scaled dimensions, and review with your local inspection office to get the approval you need before construction.

Of course this room will need it's own light with a wall switch. And insure the space around and in front of the panel will remain clear of obstructions and combustible materials.