The army of the 1st century was (according to Syme at least) the Emperor's real base of power. When Augustus died the army almost rejected Tiberius and right away he had to deal with two very serious mutinies on the German frontier. It was a very close thing, and Tiberius only got out of it because his nephews managed to talk the mutineers down.

By 69 CE Roman field commanders had figured out control of the army was more important than control of Rome, producing four emperors in one year as each successively marched on Rome. Tacitus called it "the secret of Empire" - that is, emperors are made in the provinces.

Something similar goes for the Praetorian Guard in Rome. Again under Tiberius, his prefect of the Praetorians attempted a coup based on the guard being loyal to him, which Tiberius slipped out of with some careful cloak-and-dagger work. After Caligula was assassinated, it was the Praetorians who forced the Senate to accept Claudius.

Basically, if you're emperor, you want the guys with swords on your side.

That's a pretty cynical view I just stated; in reality there was a lot of complex ideological work that went into legitimizing these regimes (including the cult of victory), but I think it's fair to say since Syme published in 1939, the cynical view is the most common one.

It was especially important for the emperors because as the empire expanded into new territory, this would lead to the requirement of maintaining garrisons on the frontier. Thus Emperors such as Augustus, saw the importance of maintaining a long-standing, professional army, and made reforms such as pay increases, stipulations of land and pay upon retirement. Also later in Roman history the army themselves would put fourth candidates for emperor.

The emperors kept support simply by going on campaigns. By the time of the empire, being a soldier was a professional career for many. Campaigning insured the prospect of booty, personal fame, and glory. Also as mentioned above, pay increases helped.