10 days in Lake Garda with teens, one base?

It’s up to you, but even though the lake is long, since you have a car, I don’t think you need to split your stay. You can drive from the top to the bottom of the lake in less than 1,5 hours.
$1,000/ night is a lot. I see you chose 4 star hotels, which are more upscale but still I doubt they will cost more than 250€ per room, so your budget is plenty. The issue might be whether those hotels have availability for those dates. You are going to the lake during the busiest time, and the lake is popular, especially with those interested in outdoors activities. The lake is a paradise for every sports activity you can think of: Windsurfing, sailing, parasailing, kayaking, hiking, biking, cliff climbing, cliff diving, gorge motorcycling, jet skiing, auto safari, wine tasting, water theme parks. The cliff diving is mostly in the north, especially Lake Ledro, a smaller lake near Riva. Also around Riva there is the Varone grottoes and waterfall as well the Ponale waterfalls. Malcesine is famous for cliff parasailing (Monte Baldo). Given the above options, staying in Riva or in the north in general, will put you closer to what you are interested in. Gardone Riviera is also not too far.

Some years ago, I stayed at Villa Florida for two weeks many times over the years. Beautiful location and apartments, run by a lovely family.
From there you can drive to anywhere on the lake, and there is also a short car ferry from nearby Maderno to Torri del Benaco on the eastern shore, as well as the longer one from riva to Desenzano. Plus, of course, the foot ferries.
Several restaurants within walking distance, plus more a few minutes drive.
As stated above, there are numerous watersports all over the lake. The winds are more challenging the further north you go.
We used to wind surf at Gargnano.
Have a great time.

Since you’ll have a car this is doable, but if any cities are on your itinerary you’ll really be too far for a daytrip IMO if you base in Riva. So depending on your flight plans, you may want to use a day or two at the end to explore Venice or wherever you’ll be departing. Just a thought for some contrast :)

If you stay at Lake Garda and have a car, consider a daytrip to one or more of the neighboring cities. You can get your outdoor activities in or around the lake and also take advantage of the ruins, museums, architecture, food and atmosphere of places like Brescia, Verona, Trento, even Bergamo or catch the train to Bolzano. I suggest splitting the ten days, devoting most of your days to your lakeside adventures and at least two days to a city destination of your. choice. You did not say if your family has visited this area before or if this will be your first trip to explore the area.

Thanks everyone! I decided that we are going to stay in Riva at Hotel Louise as our base, with a night or two in Venice at the beginning or end.

Not sure if your kids may be interested, but there is also a famous aqua park called Gardaland.
There is also a zoological auto safari park called Natura Viva. I went to that auto safari park when I went to lake Garda the very first time, like when I was maybe 12. I enjoyed it then. Make sure you don't rent a convertible or they won't let you in.