Many folks will suggest you prune at ground level, then start over.

Or you could graft those apples onto fresh rootstock.

Or, if you're convinced you want to keep the trees, check out this publication from Oregon State University.

http://extension.oregonstate.edu/catalog/pdf/ec/ec1005.pdf

That said, understand that the process requires 3 to 4 years to do it correctly.

Here is a link that might be useful: Pruning to restore an old, neglected apple tree

I would not recommend you to start at ground level.
Prune out the dead wood right now. Also cut off the vertical wood. Expect in July to go in and cut off the vertical again as it will regrow. It will not regrow much after summer pruning.
Probably more important than pruning is that you get a good cover on there for San Jose Scale. I would suggest Lorsban which will knock the devil out of them, but it is a restricted pesticide. If you can pay about $35, pass the test and purchase something like a backpack sprayer then you can buy Lorsban. It is not cheap, but it will last you a long time and it is very, very effective. I don't where you are in WA as you didn't state, but I can assure you that you have a heavy scale infestation and it will shortly kill those trees. That is very possibly what has killed the limbs that are presently dead.
I would disagree about three to four years to rejuvenate. A dormant, a summer, and another dormant will have those trees looking good. So that would be about 15 months.
You are going to have to spray them.

If you can post a picture of the trees, it would be a lot easier to reply. Some of the old 'Pioneer trees'- most in fact, are full of heart rot from improper care and many are also virus infected. When you say "old", just how old do you think they are?
How large is your property? Can you live with the apple trees using up half of your lot?
I tend to think that, unless a tree has immpressive value as an ornamental or shade, it is best in many respects to cut them about at ground level.
We now have a variety of size controlling rootstocks, that keep fruit trees below 12 feet tall. This means they take up less room in the yard
Are easier to prune
Are easier to spray
Are easier to harvest
Also, the size controlling rootstocks bring the tree into bearing age faster than a standard tree.
The cost of disposing of pruning branches on a standard tree or several can easily cost $100/year.
Are the apples on one tree enough to justify this cost?
So, in brief, unless one of those trees has outstanding fruit and has an attractive form...........
Call for the chainsaw.
Apple does make awesome firewood.

I have 2 old apple trees as well. They are from my greatgrandfather. They produce wonderful apples and great flavor. I would like to know if I need to spray them?

I have some scarring on them that I would to get rid of.

Thanks