Here's how I'd do it. The closest beach is Playa del Rey - it's quite and non-commercial, you'll have to walk for 5-10 minutes to find a coffee or an ice cream, but it's nice.
Take the free parking shuttle to The Parking Spot: http://www.theparkingspot.com/locations/locations.aspx?ID=22
That will take you to Westchester. Eat at In'n'Out, or buy snacks from Trader Joe's, or have German beer and Swiss/German food at Chalet Edelweiss (food is a touch pricey, but interesting).
Walk to the corner of Manchester and Sepulveda (about 3 blocks North, by Trader Joe's), and catch the 115 bus towards the beach. Note that not every 115 goes all the way to the beach. The timetable is here: http://www.metro.net/riding_metro/bus_overview/images/115.pdf
Hang out at the beach, and then catch the bus back. Be careful, because the buses are only about every hour. If you're in trouble, the worst case scenario is walking a mile and a bit back up the hill to Sepulveda, where you can catch the Big Blue Bus number 3 to the LAX bus center (described by @choster).