I just got this as a question, and knew the answer immediately, though wanted to double-check. Yes, you can put a baby in an approved car seat in a convertible. I know because I’ve had a BMW 3 series convertible since our kids were born (had to ditch the Z3 however), and they ended up there quite often, though it was a pain because of the single doors. Newer models all have significant roll-bar protection, but as with all car seats, make sure they are very tightly fastened. 
From what I gather, almost all states even allow a car seat up front in two-seater models if there is no back seat. You have to make doubly sure that the airbag is turned off however, since it could really hurt the baby far worse than a crash in many cases.
With the top down, don’t expect admiring looks from other drivers however. Most people will look at you like you’re a child abuser. I’d have to agree if the child isn’t old enough to clearly tell you whether they enjoy being in the wind and open air. Since my kids could express an opinion, they’ve loved having the top down, but it’s only been recently that I’ve done so with no regard to the stares of fellow motorists.
I just got this as a question, and knew the answer immediately, though wanted to double-check. Yes, you can put a baby in an approved car seat in a convertible. I know because I’ve had a BMW 3 series convertible since our kids were born (had to ditch the Z3 however), and they ended up there quite often, though it was a pain because of the single doors. Newer models all have significant roll-bar protection, but as with all car seats, make sure they are very tightly fastened.