A 1 year old likely couldn't make it to Delicate Arch walking on their own, but if you have a secure backpack and are able to carry him up hills etc for 45 minutes or so, it is definitely doable. There are some dropoffs near the end and at the arch itself, so you'll need to have a close watch/hand when he's not in the carrier.

In the Devils Garden area, there are some nice short hikes on safe trails that go to some of the arches (up to Landscape Arch; you won't want to climb beyond that). Sand Dune arch is supposed to be a nice one for kids, fairly close to the road and interesting to see/run around/play at.

Canyonlands has amazing views right from the parking lots in several places; you don't have to hike far if you don't want to. A 1/2 mile roundtrip hike in Canyonlands gets you to Mesa Arch, worth seeing. Again, a backpack carrier would work well, but the trail itself doesn't have major dropoffs or elevation gain (watch out near the arch itself though).

Dead Horse Point is another worthy stop for the views, not sure what non-cliff hiking is available though.