I saw some really negative reviews about the south of France (restaurants, facilities, car hire, everything!) before I went and have joined Mumsnet today specifically to set the record straight.
Mougins, Cannes and the south of France are beautiful, friendly, full of things to do and perfectly well set up for babies and toddlers. We went at EASTER which was an excellent time to go, as it was 18-22 degrees all week and warm enough to run about on the beach and paddle without getting sunburnt. It rained once for an hour. The sea was a bit cool, but our toddler just wanted to dig and shout at the sea birds.
The Cote d'Azur was full of activities as well as the beaches and everywhere we went there was a little playground with fresh water on tap and good fences all round. The slides were downright impressive, as they have low, wide slides for toddlers as well as big-boy slides. Not all carousels were open at Easter, but some were and we enjoyed them greatly. The Ocean Museum of Monaco was brilliant, although Monaco was otherwise uninteresting for a child. Parc Phoenix in Nice was super cheap at 3 euros each and full of animals and birds to look at.
We stayed in Mougins, 15 minutes drive from Cannes, and it had a lovely old town with restaurants. The waiters were kind to our boy and not too busy at that time of year. They were happy to get him tomato pasta and bread and make a fuss of him. Every restaurant we went to had a high chair. Nowhere had a baby change room, but we managed fine on the bathroom floor.
As to money, we rented a one bed flat in modern accommodation, which suited all of our needs and was very safe. Look at the Owners Direct website FR19733. It was extremely good value for money, as were the BA flights which I booked two months in advance. BA gave the toddler a full baggage allowance 23kg plus buggy within the price, even though he sat on my knee! The car hire company provided a good quality child seat for very little extra.
In summary, if you plan properly and go at the right time of year (not July and August) the Cote d'Azur is an excellent place to take a small child, and probably cheaper than some of the more obvious choices.