There is no reason at ALL to even think of putting your dog to sleep because of your neighbors dogs!!!
As stated above, a good fence would be a great place to start. I'm sure the first thing you would think of is the cost of putting up a fence. But take into consideration the cost of possible vets bills you would have in the future without the fence.
I have mean neighbors of my own, their dogs and mine have gotten into fights over and under my fence many times. Both of my dogs have been to the vets because of bites. My hubby and I have taken extra precautions to make sure the can't get to one another anymore.
I made calls to the dog warden and sheriff department, they received one warning from the dog warden then they started getting fines. After about a year of this they finally put up a fence!!