Sparta had ceased to be the old, familiar Sparta long before the Romans arrived. The Spartan way of life began to decline sharply after the battle of Leuktra in 371, when the Thebans defeated them and permanently ended their ambitions of Greek hegemony. During the 4th century, there were periods of Spartan obstinance and brief resurgence, but the fight against Antipater in 331 all but ended them. They lost thousands of full Spartan citizen males in battle and were never able to replenish their numbers due to their own...Spartan...ideas of reproduction. The city-state remained a regional power and still fielded small, if fearsome, forces, often in a quasi-mercenary role.

In the next two centuries, Sparta remained on the sidelines of history for the most part, quietly continuing her way of life as best as possible. During the Second Macedonian War, Sparta became the hegemon of Argos. The two were ancient enemies and this must have been fun, if only briefly. In 195 the Romans used Sparta's occupation as pretext to begin the Lakonian war, resulting in the loss of Argos and the besieging of Sparta itself. Soon after, the last Spartan king (Nabis) died and Sparta was finally forced to join the Achaean League.

50 years later, Sparta had no real contribution to the Achean War between the Achaean League and the Romans. Upon Rome's victory in 146 and subsequent annexation of most of Greece, Sparta was made a free city once more, though now very much a shadow of her former self. The Spartans still lived the Spartan way of life, but now as a sort of historical reenactment of what they had been. There is not a lot of archaeology to speak of at the site, since the Spartans had never been big on monumental architecture, but we get the sense that the population was much reduced. Roman elites would come to visit Sparta as tourists over the next 200 years, probably to watch "Spartan" hoplite demonstrations which were a sad mockery of their subject matter.

I did a few searches and can find no mention of Spartans ever being used in Roman armies as auxiliaries in any capacity.