As far as I know, there is a general consensus that oliganthropia (the steady decline in the number of full Spartan citizens) was a real and serious problem in Classical Sparta. While the numbers of Spartan citizens we get in the sources are sometimes difficult to interpret, few now stand by Lazenby's inflated figures (presented in his The Spartan Army (1985)), and even he accepted the general trend. The following totals are uncontroversial:

• Late 6th century BC: 8,000
• Battle of Plataia (479 BC): 5,000
• Before the battle of Leuktra (371 BC): 1,500-1,200
• After the battle of Leuktra (371 BC): 1,100-800
• Late 3rd century BC: 100

It is clear that the decline throughout the Classical period was constant and severe.

As for the impact it had, most scholars agree that it does not appear to have affected the combat abilities of the Spartan army. Since the main Spartan tactical advantage was its formation drill, which could only be effectively practiced with the whole levy present (including perioikoi and hypomeiones), it did not matter how many men in the army were full citizens who had gone through the Spartan upbringing. Army training only really begun when the army was already on campaign.

It did, however, seriously reduce the number and scale of the conflicts the Spartans could effectively commit to without overreliance on mercenaries and allies. It was increasingly difficult for them to maintain a strong presence abroad while keeping their subordinate allies and their helot population in line. Xenophon describes the atmosphere in Sparta at the time of the conspiracy of Kinadon (397 BC):

The informer said that Kinadon had taken him to the edge of the marketplace and directed him to count how many Spartiatai there were. “And I,” he said, “after counting king and ephors and elders and about forty others, asked 'Why, Kinadon, did you tell me to count these men?' And he replied: 'Believe, that these men are your enemies, and that all the others who are in the marketplace, more than four thousand in number, are your allies.'” (...)

Whenever among these classes [the helots and the neodamodeis and the hypomeiones and the perioikoi] any mention was made of Spartiatai, no one was able to conceal the fact that he would be glad to eat them raw.

-- Xen. Hell. 3.3.5-6

This particular conspiracy was nipped in the bud, but the Spartans lived in fear of other plots and rebellions. It was oliganthropia itself that created unrest; many of the dissatisfied elements in Spartan society were those who had lost citizenship status due to impoverishment or those who felt restricted by their inability to gain further rights. The decline in citizen numbers simultaneously worsened the problem and decreased the effectiveness of any countermeasures.

When Theban successes stripped the Spartans of their allies and they relied on themselves to defend Sparta, their numbers were pitiful, and they relied on sheer reckless bravery to drive the Thebans off. The liberation of Messenia likely led to another large number of Spartiates losing citizenship status. By that time the Spartans were no longer able to claim the status of major player in Greek affairs.

Edit: Glossary of Terms

• Perioikoi: "dwellers-around" - free-born Lakedaimonians and resident foreigners who did not have citizen rights but were liable for military service.
• Hypomeiones: "inferiors" - former Spartiates who had lost citizenship when they fell below the property treshhold.
• Neodamodeis: "new people" - former helots who had been freed as a reward for military service.
• Helots - indentured population of Lakonia and Messenia who worked the land for their Spartan masters.