I love the idea someone posted about taking your child to concerts and using CDs and DVDs to reinforce music all the time. This is such a vital step. Violin or piano are performing arts, so the kids need to hear and especially see performances. Go for live ones as much as you can. Look for high school orchestras if professional is too expensive for you, and look for listings for recitals -- often, churches or community centers host recitals.
As for helping, you must attend the lessons, all of them. Don't hover; if there's a way you can observe and listen without being directly in the room, do that. Your son doesn't need you leaning over his shoulder (and the teacher may not want that either) but you do need to hear what the teacher says and see what the teacher means when he or she talks about correct fingering, posture, etc. Be sure you know each week what the "homework" looks like and sounds like -- which measures of music your child should be working with before the next lesson. Even if you don't read music you can, with the teacher's help, at least see what portions your son should be playing. Work with that teacher and explain that you want to help without pressuring, and want to get advice on how to do that. You do need to know what proper practicing for each week should sound like -- what is the piece that he's working on, and what does that sound like properly? The teacher should play it for him at the start of a new piece and you should listen. At the end of each lesson, spend a few minutes with your son and the teache and have the teacher go over the week's work. Have your son write it in a small notebook to keep with his music at home. You don't mention whether you have a piano at home but I assume you do, or will get one if he takes lessons. Not having one isn't really an option.
Practice has to be daily, with maybe an occasional skip on a tough homework night, but too many skips add up to trouble. There's "muscle memory" involved and kids won't develop that unless they are consistent. Ten to fifteen minutes a day likely is fine for a beginner -- if the teacher wants an hour a day, you have the wrong teacher at that level.
If you need to use a reward system for practicing, go for it. If he can practice at a time that's not involved with homework or after-school activities, that's best; we do it in the mornings before school (but then, we have a late school start time).
How old is your younger child? Don't start him too soon. Some programs promise success for small kids of four, five or six but many teachers refuse to take kids that young. He must be able to concentrate and focus for enough time for the lesson and to practice. I would wait until he's at least seven to start piano. Get references for teachers from other parents who've used them; you want someone who is used to working with kids.
Regarding your older child and violin -- if he's in elementary school that's often when violin and other strings are offered through the school. He may be more willing to practice if he tries that. My daughter must do at least 90 minutes a week to get an A in violin lessons at school and that A really motivates her. Also, in school they are told they must commit to a year of strings, and sign a "contract" to that effect; that might motivate him too.
Violin is tough; it's much harder to make a good sound the first time out on a violin than on a piano. Anyone can touch piano keys and make decent sounds that seem like music, but the mechanics of playing violin mean that initial tries are often screechy, and that puts kids off and discourages them. It IS doable to play violin as a kid if the instruction is good, but even your older son may have more success earlier with the piano. Piano is a good foundational instrument if he wants to go on and learn other instruments.