I've lived in the Upper Valley since 1985. I attended Lebanon High School, and moved back there after college. When it came time to buy a home, we moved to Windsor VT, just 1 exit down 91, because houses were so much cheaper. My wife works for the College in Hanover NH, I am self employed in VT, and before that worked for companies in Vermont.
You will find little practical difference between the towns on either side of the river when it comes to drugs, or general social issues. You won't "avoid" drug issues by living in a NH town that is 1 mile from theVT border.
This area very much straddles the border and you'll live and work with folks who have lives on both sides. Your kids will play sports against teams in both states.
Towns like Hanover NH and Norwich VT are generally on the "upper crust" end of the scale: expensive homes, great school (though not without it's own problems) near-zero poverty rates.
Towns like White River and Windsor are still carrying largely our of date "bad raps" on the lower end of the scale as they've clawed their way back to respectability from their post-factory closing years. Every town has it's positives and negatives, it's all about finding the right place for you and your family.
Property taxes vary from town to town. VT has sales tax, but NH has other fees that make sure they get the money out of you one way or another. I haven't found an appreciable difference cost-wise from one to the other. Both have their
financial positives and negatives
If you have kids or are planning on having kids, pay attention to the school systems, but most are quite good (especially by national standards.) Some of the smaller towns like Harland VT are sending schools so the kids have a choice of High Schools. That can create travel issues, but opens up some awesome choices, so make sure you know what you are getting into.
This area is made up of a bunch of small towns, with most of the jobs centrally located in and around Hanover and Lebanon NH thanks to the College and the Hospital.
We live in Windsor and we like it a lot. It used to have a pretty bad rep but it's become quite a nice place to live. The schools have improved tremendously over the last 10 years and the town itself has been on a real tear with grants and community investment. There aren't a ton of jobs, but it's an easy 20 minute commute to White River and the rest of the UV.
Honestly, I suggest renting for a year while you learn the area and I bet you'll find the right town to put down roots in. The only place I really don't suggest is Claremont because it has the highest property taxes in the state + the worst schools + some of the more serious social problems (teen pregnancy, drugs) around.