First, take a good look at them -- how are they breathing while they eat ? I had a 5 yr old that had constant sore throats that the DR said were viral, so they weren't ever treated. They were really an allergic response -- and while she didn't generally chew with her mouth open, the reality is that there were times when her sinuses were plugged enough that she needed to breathe through her mouth. so, just be aware of the causes.
I think you should try to make this fun. You can harp all you want, and I'm sure my mom did, in order to get us to eat nicely. But what we did when my older girls were little was to have periodic "fancy meals" -- jointly with my sister's family. Everyone dressed up and we ate at one house or the other. And everyone practiced good table manners. You could call them princess meals or something. Let the girls dress up, and make sure the food is something they like, then you dress up, too, and all of you practice good table manners. Let them know what you expect, (and if they watch the Princess Diaries, or something, maybe they could provide imput into the expectations )and let them know that if you put your elbows on the table, or talk with your mouth full, that they should notice and remind you, too. And don't get insulted when they do. It's important for all of us to be reminded. I know I've gotten sloppier about it as I've gotten older. And sometimes I think we should reinstate the fancy meals at home !! (to train me, as much as the kids !!)
If you do this, you'll find that the kids tell you that it was a memory they smile over as adults. Rituals build family cohesion, and it's just one of those things that was "special" about living in your household. When my adult girls (I have two sets -- 2 adults and 2 teens now) talk about the past, it's one of their fond memories.
the other option is to simply have them try breathing thru their noses while they chew. Don't yell at them about it initially, but ask them if they can try it -- as if it's a challenge, and respond appropriately when they accomplish it. they probably don't think about it. But once you know they can do it, and they know they can do it -- you can make it fun -- I bet you can't . . . . get through two bites without chewing with your mouth open . . etc. And provide something special -- like dessert, if they win !! The more fun you make it, the more enjoyable for them and the faster it'll get accomplished. :-)