Try opening a Cmd window and running your install's sqldeveloper\sqldeveloper\bin\sqldeveloper.exe from there. Look for any crash related messages in the console and post them back here.
Edit: Oh, I assumed by "it shuts down" you meant crashes, but perhaps are really do see a shutting down dialog?
Hi, you provide very little detail in your java configuration which developer runs under. What version are you using, did you download Windows 64-bit with JDK 8 included? You could also go to your control panel, under java - advanced then show console, you might see an error written to the console when it crashes.