Start off by looking up the most modestly priced hotel that you are willing to stay at, and email your boss the link with a message like:

> This is the cheapest reasonable accommodation I could find, but you only budgeted \$#.## for accommodations.

Don't bring up that there are cheaper options unless he asks.  Don't define "reasonable accommodation" unless he asks.  Don't even ask him to take a specific course of action unless he asks what you would prefer to do.  You are simply giving him the best option you find reasonable and enough information to let him know that it does not fit into the budget.  This places any outcome squarely on his shoulders.

**In many cases this will sidestep the whole dilemma:**

Either, he approves a higher budget and you go (this is more likely than you may think), or he decides it's too much and you don't go.  If HE says you don't go, now it's his call instead of yours so you have not done anything against his wishes.

**How this improves outcomes where you still have to address the issue:**

If he asks you why you can't find anything cheaper, then you explain that everything else you found for cheaper were hostiles and in high crime rate areas.  While this sounds like it just puts you in the same predicament you're in now, it's not.  Now he's asking you a directed question that you are just responding too.  It comes off as less needy when your grievances are an answer to a directed question or statement, than when its something you bring up yourself.