You should shoot them down with a slingshot (Fandom):
Presents can also appear attached to balloons and must (...) shot with a slingshot.
Do note though, as further down it states:
If shot down over water or flowers, the present will be destroyed and the player will become shocked by it. The present can also be destroyed if it lands in the center of a 3x3 pattern layout.
Fandom also has a nice tutorial/strategy in case you do not have a slingshot:
If the player does not have a slingshot, there's still a way to knock the presents down from balloons. First the player should find a balloon. If it's drifting towards a cliff (depending on the layout of the town, there will be a cliff to the east or the west) he/she is in luck. Wait for the balloon to reach the cliff.
As it passes the cliff, the balloon will lower. The player should get within range of it with a net or shovel out, and use the net or shovel as if he/she was hitting a villager. The balloon should pop, and the present will drop to the beach below. If the player misses, he/she will have a moment or two to try again.
This strategy will not always work, as the balloon may be heading in the direction away from the cliff. If so, the player will have to wait and try again with a balloon heading the other way.