Basswood is usually used for carving. It has little grain characteristics and is very bland. It is, IMHO, too soft for trim.

Basswood is a great source of wood used when making interior window shutters as well as constructing drafting tables, broom handles, carvings, turnings, furniture, mouldings, millwork, musical instruments, woodenware, food containers, and surfboards.

Another suggestion is yellow poplar. Poplar stains very similar to pine and it might give you the look you want. It's not as soft as basswood.