Can crew members start with experience?

Solution 1:

No, none of the crew that can be acquired in FTL as of this version start with any experience. However, it has been a suggested feature on the FTL forums here.

Solution 2:

As of the Advanced Edition, at least, it is possible - however generally all EXP bars will be empty until you reach later sectors, and even then they will have a "speciality", ie. a single stat with EXP with the others all empty (which is how my crew generally ends up anyway).

This can apply to new crew members received from events too, as will be shown by the corresponding icon next to the crew member in the message telling you about them joining you.