In a double-sided drive, there are two heads which are necessarily closely spaced, facing both sides of the thin magnetic-coated disc once it is inserted. The concern would have been that the heads could collide with each other if the drive was jarred.
However, the drive designs I'm familiar with have a mechanism to hold the heads and drive spindle away from the disc until the drive door is closed (or, for the 3" and 3.5" stiffies, until the hard casing is detected in the inserted position). This is to permit the disc to be inserted by an untrained user without damaging the disc, but it would also hold the heads apart from each other so that they could not collide.
If the drive were jarred with a disc inserted, the heads would collide with the disc surface - but that's also true of a single-sided drive. I think floppy discs are a bit more tolerant of that than hard drives, mainly due to the lower speeds involved.
In practice, double-sided drives became common and did not have a reputation for fragility. I have had several laptops (and even an 8088-based Amstrad luggable) with 3.5" double-sided drives, some of which had decidedly rough handling over the years but gave no trouble.