Be aware that there is still a LOT of snow in and around Silverton. Last I heard, the San Juan Mountains were averaging 700% of normal snow pack! So while the town itself may be melted out, many of the trails, roads, etc. will not be easily accessible. Postholing a hiking trail isn't something most kids will enjoy.
South Mineral is in the trees so may still be filled with snowdrifts, even if it is open. At Molas Pass, the highway is surround by deep snow drifts, so Molas Lake might also be a little rough.
Maybe consider camping around Durango. You can still day trip into Silverton, or better yet, take the kids on the Durango-Silverton Narrow Gauge Railway. One of the finest (and "funnest") things you can do in the area. Your kids will remember riding behind a steam locomotive above the river canyon with amazing mountains all around. Also from the Durango area, you could do a day trip to Mesa Verde. Get reservations for a guided tour of one of the cliff dwellings. Yes, you have many NM with natve ruins in your area, but these are pretty special. Rafting the Animas River should also be good right now.
Near Durango: Of the 43 campsites at Haviland Lake, 20 miles north of Durango, 26 are reservable meaning 17 are first come, first served. Junction Creek CG is around five miles NW of Durango and still had reservable sites mid-week plus there are 15 first come, first served. If these two are full, check Target Tree, 20 miles east of town and centrally located between Durango and Mesa Verde.
We've stayed many times at all these CG's around Durango and Silverton. Given the snow pack right now and your arrival time, your best bet may be Junction Creek CG out of Durango. Haviland Lake is nicer, but there are no reservable campsites left and arriving after noon, even on a Monday, would be dicey. Plus you can still get a reservation at Junction Creek so you don't have to rush to get there.
