TBH I'd say most of Ibiza is pretty family friendly. There are loads of beaches, ranging from tiny to really quite large..
Playa den Bossa does have several clubs and is definitely party central but they co-exist with lots of family hotels. I've stayed there with children several times. I'd probably try not to be near Ushuaia as their music is open air.
The beach there is quite large and good sand. It is on the outskirts of Ibiza town so can be quite busy, but from most hotels you can walk to the beach easily.
Cala Llonga is a favourite of ours and is very family friendly, nice sheltered cove with good sand, loos, showers, lots of beach umbrellas and a couple of small supermarkets and restaurants. Much lower key than Playa den Bossa.
Will you have a car? Driving is easy and the whole island is only about 25 miles long. With a two year old I would probably base myself at Cala Llonga and potter up and down the coast, but if no car I'd definitely go for Playa den Bossa because it is easy to walk into town.
You might also want to look at Santa Eulalia which is a small town and beach resort which is family friendly and has a bit more going on than Cala Llonga. Handy for a trip to the Hippy Market at Punta Arabi too.
Happy to give beach suggestions if you do want to drive around, I've been going there since my children (now teenagers) were quite small and it is still our preferred holiday destination.