So far the answers you have received are pretty accurate. There are no restraints in the bassinets because if turbulence occurs (or the seatbelt sign illuminates because turbulence is anticipated) you must remove your child from the bassinet. No airline would provide a restraint or encourage you to leave your child in a bassinet because for the following reasons.

They are only held in by three push pins that lightly lock into the wall. Who would trust the integrity of these pins over the strength and security that will be found in a paternal embrace? These bassinets are only weight listed to a maximum of twenty pounds. A ten pound infant can exert a twenty pound plus force if the aircraft is seesawing side to side or moving violently up and down (or can be catapulted from the buoyancy obtained in some situations). That is why infants must be removed when we start jerking you around

If you do not go the route of a car seat, as suggested, hold your infant in the burp position. You will be in a more secure position to react protectively and quickly. In a split second, if you feel your child is in danger of becoming airborne, you can instinctively tighten you grip to prevent this and the burp position has proven to leave the parent in better control to cover every direction that your child could be launched from. This same advice is given by way of our training in a planned emergency. You, and the shield your body provides, is the best protection your child can get.

Your love is the best defense and protection your offspring have and this has been proven over and over.