B&J, the other replies are spot on.
I am not currently living in Costa Rica but have lived there off and on before and plan to move there "permanently" (or so we think) next year.
Yes, crime has increased in Costa Rica. The biggest problem is in or around San Jose' and for the most part outside San Jose' muggings and so on are not common.
What IS somewhat common - depending on where you live - is burglary when you are not home. I believe this can be minimized with good security such as a real guard dog (big enough and trained to do the job), a sign that says you shoot burglars (and back it up with a gun that your neighbors know you practice with), lights that come on with movement and trying not to leave your home alone too much in the first place. Also you can put any portable valuables in a safe closet built to NOT be broken into (this is if you build a home, which you said you would not). In that case perhaps a big safe to put stuff in or some hidden compartments etc.
So burglary is the #1 crime you have to worry about.
Rent:
It can be found very reasonable if you ask around (assuming you don't want to live in a Tico home with less security, perhaps bad plumbing -suicide shower, can't flush toilet paper-, poor water pressure etc). I have on several occasions found very good deals on American built homes to rent, my most recent one being a 3 bedroom brand new home that was going for $600/month, with security, beautiful views, etc. (It's no longer available by the way...) But by spending some time looking around and asking around - check with realtors, check with everyone you run into - you may find an even better deal. 6-7 years ago I knew of some really awesome deals in the Puriscal area. But yeah, they've probably gone up. Double? Maybe, maybe not.
Panama?
Let me give you the down side of Panama:
More infrastructure: Yes. But also federales everywhere, constantly stopping people randomly or at regularly spaced booths on the highway. I don't do anything illegal yet it bugs me to be stopped by federales all the time.
While Panama has tons of forest parks etc., the fact is at least where I've been, you don't really SEE the nature like you do in Costa Rica. In Costa Rica you see nature everywhere you go, all along the roads and highways. In Panama it's like you see it in the distance and have to make a point to "go to" a forest, more like in most parts of the USA. Are there parts of Panama more like Costa Rica? I am hoping that is true but I did not see them in the northern area I was in for several days.
All I can really say is that my friend and I went to Panama and got bribed by a federal border agent who threatened us with jail and his gun. That took up one of our days in Panama - it's a long story. (Short version: We made a mistake at the border by not getting a stamp we needed and we had to make a trip back to the border and on the way back a federal agent got in the cab with us - knew we were coming so watched for us at one of those road blocks I mentioned) and made all kinds of threats to our safety until we paid him off. An anomaly? Maybe, but that was my one trip to Panama so I don't know. It did give me a bad taste in my mouth for Panama. I've NEVER been bribed or in any way hassled by a policeman of any type in Costa Rica, with visits and living periods there, over 20 years.)
My friend and I also just felt that Panama was not as tranquilo or easy going in general ((aside from the run in with the federal border agent).
Now, am I saying don't move to Panama? No.
I am actually looking forward to spending more time there, to give it a 2nd chance, and were I to find an area I liked, an area with a lot of nature, more like Costa Rica, I will consider moving there.
Panama IS cheaper! We stayed in an amazingly nice hotel in David for like $40/night. Reminded me of how Mexico City used to be, cheap hotels and cheap food. I have a friend from there though and he confirms the police presence, the police bribes, and also told me there is a sizable amount of anti-American sentiment. He actually was born there and grew up there so he knows what he's talking about. Again, that said, I want to take another look at it, next time spending maybe 2-4 weeks there.