Assuming it is a guaranteed connection, depending on the length of delay, Amtrak will either hold the connecting bus (and sometimes train), arrange alternate transportation to the destination (or to catch up with the missed connection), or place a passenger into a lodging accommodation with a meal allowance and transportation on the next train out (usually the next day).
On the Southbound Coast Starlight I know when we were delayed due to a fatality involving train 14 in Santa Barbara the people who had to connect to the Sunset Limited were off-loaded in Goleta (non-scheduled stop) and placed into Taxi's to LA to make the connection. Many people going nearby (including myself) were also allowed to get off in Goleta and end our journey there (and have the people meeting us pick us up there or what not). The southbound Surfliner connection at LAX to San Diego was held for train 14.
That delay was quite substantial (train 14 was delayed about 6 hours, train 11 about 1.5 hours, and Surfliner trains were involved in the back-up as well. The northbound Surfliner was attached to the rear of train 14 and the southbound Surfliners were coupled together to operate as one train.)
All and all if you are moving now, depending on your train speed, there is recovery time built into the schedule as well between SBA and LAX. It won't make your train close to on-time but it will help a little bit (up to an hour at most).
Good luck with your journey! It's foggy here in the Bay Area but hopefully there will be some time left to see the sun set as you journey along the coast. If not you'll see the lights from the oil rigs out in the Ocean. It's still a nice ride though.
