How secure is an Azure database within Microsoft?
The only way to be secure, is to control your data. If someone else is running it on its own infrastructure, they are able to read, change or use them. Period. 
For example: if your database is confidential one, you should not use someone else infrastructure.
Saying that ( which is completely obvious), you may assume, that as long as you not breaking the law, and USA is not in political conflict with your country, Microsoft is not interested in peeking at your data, exactly as any other company

So if you have something to hide, use your own server. If you have nothing special ( and putting it on the web is in compliance with local law) go on.