James Fernald, English Synonyms and Antonyms, twenty-first edition (1914) lists debase and degrade in a synonym group that also includes abase, bring low, cast down, depress, discredit, disgrace, dishonor, humble, humiliate, lower, reduce, and sink. Fernald offers this distinction between the two words:
Debase applies to quality or character. The coinage is debased by excess of alloy; the man, by vice. ... Degrade may refer to either station or character. An officer is degraded by being reduced to the ranks [of common soldiers], disgraced by cowardice; vile practises degrade; drunkenness is a degrading vice. Misfortune or injustice may abase the good; nothing but their own ill-doing can debase or disgrace them.
Similarly, Webster's Dictionary of Synonyms (1942) identifies debase and degrade as members of a synonym group that also includes abase, demean, humble, and humiliate, and offers this commentary on the two words:
Abase, demean, debase, degrade, humble, humiliate are synonymous when they denote to lower (one) in one's own estimation or in that of others. ... Debase emphasizes deterioration in value or quality it is more often used of things (as , to debase the currency), but when used of persons it commonly connotes weakening of moral standards or of the moral character; as, officeholders debase themselves by accepting bribes. "How books ... debase The many for the pleasure of those few" (Wordsworth). Degrade stresses a lowering in plane rather than in rank and often conveys a strong implication of the shamefulness of the condition to which the person (or group or thing) has been reduced. "That she and Charlotte, two spent old women, should be ... talking to each other of hatred, seemed unimaginably hideous and degrading" (E. Wharton) Often (esp. in degradation) it connotes actual degeneracy or corruption. "It was by that unscrupulous person's liquor her husband had been degraded" (Hardy).
Much like Webster's, S.I. Hayakawa, Choose the Right Word: A Modern Guide to Synonyms (1968) links debase and degrade to the words abase, demean, disgrace, humble, and humiliate:
These words indicate the act of stripping someone of pride, self-respect, rank, or reputation. ... Degrade may or may not refer to a literal demotion: a sergeant degraded to the rank of private first class. More often the word suggests disgusting or immoral habits or behavior that destroys a person's character or publicly disgraces him: a fine mind degraded and dulled by dope addiction; accusations intended to degrade him and destroy his reputation. ...
Debase suggests an action that reduces the intrinsic value of something: currency debased by inflationary measures; political discussion debased by an atmosphere of hysteria. As can be seen, debase may be more factual or descriptive than degrade, which suggests a more thorough but more subjective or moral sort of depredation.
I think that the last sentence in the Hayakawa excerpt is far more on point than any effort to impute a basic division in meaning that applies debase to things and degrade to people. Debasement is certainly a lowering, but it may be asserted at the end of a dispassionate analysis. Degradation, on the other hand, more often has a moral dimension, including a reaction of moral repugnance.
Still, I advise against treating these two words as if they were distinct in compass when properly used. In reality, the two words have a tremendous amount of overlap and even interchangeability—as do they and allied words such as demean, discredit, and disgrace—in everyday speech. The reference works that I've cited here do a nice job of identifying trends and tendencies in historical usage (and to some extent in contemporary usage), but they point to nuances, not to fundamental differences in meaning or usage.