Short message, mention the gift, give a little anecdote re baby if at all possible (ideally involving the gift "ds is asleep right now in the baby grow you sent", "dd has merrily pooed in most of the nappies already" etc!)
sign from you and DH...don't sign on behalf of the baby (imho)
I think you definitely need to write a message, so that you thank people for what they gave you - especially if anything were sent straight from the supplier (so eg if they paid for a bale of 24 nappies, but you only received 12, they know to take it up with the trader). Also, though most people are far too polite to ever let it show, it is a bit disappointing to think your present made so little impact that you had forgotten what it was even before the thanks were written.
Of course, if anything sent was truly bizarre or loathsome, then you'll have to bring out your best creative writing skills so you're true reaction isn't apparent ....
I would be tempted to use them in a similar way to a standard Thank You card, ie, mention the gift specifically. One of you can sign for both of you, I think its more important that people their gift is appreciated. Hth.