I was present at our dog's euthanasia when I was 3 or 4. My kids were 3 and 6 when they had to face their first euthanasia. I asked if they wanted to be there. They did, so they were.
I think it's important to let kids be there if they want to be. It shows them that it's peaceful, not scary. Also, avoid the term "put to sleep". Be gentle but blunt. "Fluffy is very old and very sick. The doctors can't make her better anymore and she hurts. The vet is going to give her a special shot for animals that will make her die. It won't hurt her at all even though we will be very sad." Or some variation thereof.
There is a book called Dog Heaven that is wonderful for kids if that's what you believe and teach. If not, that's ok, just focus on the fact that she won't hurt anymore. Talk about the best moments with your dog. Talk about the funniest thing she ever did, the naughtiest thing she did as a puppy, etc. Laugh about the good moments. Let your daughter help spoil the dog rotten. Get her her own Happy Meal from McDonald's, make her a steak, whatever forbidden things she loves.
As a vet tech: whether or not your daughter is present (but especially if she is), make sure your vet does an IV catheter and sedation before the euthanasia. An uncommon but not rare reaction to the euthanasia shot can appear very violent and is very traumatizing for owners. The pet is gone and doesn't feel a thing, but it's awful for an owner to see. An IV catheter and sedation ensure that won't happen and it will be as peaceful for you as it is for your dog.