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Do they refer to a "left" and "right" bank in Geneva? In any case, it doesn't matter. You want as central as reasonably possible. Use these parameters; no further out than the train station or the Hotel Wilson. Four Seasons and Hotel de la Paix are not in your budget but those are ideally located so look around those two.

On the south side of the river, there are a few very reasonably priced hotels but they involve somewhat of a walk or tram ride (Pax).

Our son's Bar Mitzvah was held in Geneva so we stayed in a number of hotels during our many visits, as well as got reports from the guests. The Hotel Auteuil had really small rooms. I went to visit a girlfriend and her family there and they complained about the staff. But it looked clean and nice. Not sure how much better it would have been if they spoke French. It is on a busy street and doesn't have a view.

No one I knew of stayed in the Jade but my husband's cousin stayed in the Edelweiss, another Manotel like the Kipling, and they liked it very much. They were a family with two small children. EastWest gets good reviews but again, no one from our party stayed there.

A bunch of us stayed at the Warwick, which I can recommend. The higher floors sort of have a view and it's in a convenient location. The rooms were bigger than those in a lot of other hotels.

Be aware that very few hotel rooms can hold a foldaway bed. You may also look into apartment/hotels. My in-laws stayed in one, it was okay but well located but I can't remember the name. Alternate accommodation might suit a family better.

I trust Tripadvisor but unless you can find a review from a family, it's difficult to tell which hotels are best for families with young children.