How should I boil green beans to keep them crisp?
You shouldn't cook more than a few minutes, and should cool them as quickly as possible afterwards.  Ideally, you'll be blanching them, and here's how:


Bring a pot of water with a pinch of salt in it to a roiling boil
Dump beans in, and cook for a few minutes
Check that beans are fully cooked (time will vary by variety and ripeness of beans)
Strain beans, and IMMEDIATELY transfer to an prepared ice water bath to cool as fast as possible


This ensures beans are precisely cooked, but do not have a chance to over-cook, and using the ice water helps them retain color and flavor. 

It's how we get deliciously crisp haricot vert (French green beans) at the restaurant I work for. Note that cooking times and crispness will also vary based on the variety and age at which the beans were picked.