Multiplayer radically enhances your chances of getting the best possible ending. Basically, you have a 'Militry Effectiveness' score which must be over 5,000 for you to get the best ending. Your 'Galaxy at War' multiplyer effects this. If you do no MP at all, your GaW modifier remains at 50% (i.e. to get 5,000 effectiveness, you actually need to get 10,000).

The problem is that by doing every single side-quest, collecting every single War Asset on the map and getting the best outcomes from each side and main quest, apparently you can only get to 6-7,000 effectiveness. With no MP, that drops to 3-3,500. You CAN still get the 'best' ending, but only if you've imported a ME1/ME2 character (preferably a strong Paragon one).