It usually takes 40 to 45 minutes, door to door. In the late afternoon, you can get stuck in traffic leaving Baton Rouge.
It might help to see one of the new Google Maps hybrid views, showing Livingston, I-12, LA-63 and LIGO .
If you have Google Earth, here is a kmz file that will zoom you into the LIGO Livingston corner station.

Starting at Nicholson Hall on Tower Drive in the LSU Campus, head South.
Turn left on S. Stadium Drive, which flows into W. Lakeshore Dr.
Turn left again on Stanford Avenue.
Continue on Stanford until the intersection with I-10, and take the ramp on the right marked East.
I-10 East splits into I-12 and I-10. Bear left, marked I-12 East.
Drive 22 miles to exit 22 (Livingston/ Frost), and turn left at the end of the ramp; you are now on LA-63 North.
Turn left on Florida Blvd (rt 190), and then right on LA-63 North after a short distance.
Turn left at the LIGO sign (made of anodized aluminium, designed in Beverly Hills). There are those who call this road LIGO Lane.