I have one 3 yo who is a nose picker and the other 3 yo is the opposite...lets her nose run until it's in her mouth....blech!!
With the nose picker, I always ask him to get a kleenex when I see it happen. I remind him that it's gross and it spreads germs on our toys. When a toy gets dirty here (usually from them putting in their mouth) they have to put it in the sink and I will wash it the next time I do dishes. So if all the toys are in the sink they won't have much fun! He sometimes gets nose bleeds from picking so I remind him that he doesn't want to get "sick" and have his nose bleed because if he does, he will have to sit down in the kitchen until it stops and he won't be able to play with his friends.
With the runny nose, I remind her she is a big girl and she doesn't need to wait for me to wipe it. I also have kleenex where they can get at it and she knows that she is able to go and get one whenever. I also remind her that she will get sick if she keeps eating her boogers and if she's sick she can't come to daycare.
It's a never ending battle with both of them lol. Hopefully it's a phase that they all age out of soon!!