I recently tightened a supply line nut on a faucet. I didn't check the overall connection and a leak to the room below alerted me to the fact that I had twisted the rigid decorative supply pipe just enough to break the compression seal on the lower end.
If yours is a new installation, it is likely you have a flexible supply line, so they are probably not at risk unless the line is pulled taut. If the line is rigid (e.g., copper pipe), this would be much more prone to leaks based on even small shifts.
Glued connections on PVC pipe are pretty strong and probably pretty resistant to very small movement, but they are intended to be a seal and just structural enough to support their own weight. I think your caution is well placed. What about tying the back into the studs in the wall it sits on (or even drywall with toggles)? If there is no wall. What about building a frame around the base? This would block the drawer shown. Even boards along the sides and back, attached to the floor with screws, would help minimize lateral movement.
SUPPLEMENT PER COMMENT: There are new toggles where the bolt can be readily removed and reattached:
The metal internal brace is rotated parallel to the plastic strips and inserted through the hole in the wall. The strips are slid until the metal brace is perpendicular again. The outer flange is slid down the retaining strips until it is flush with the wall. The excess strips are broken off. The bolt is inserted separately and can repeatedly be removed.