Spousal privileges stemmed many eras ago from a romantic view of family as an institution. The theory was that we should revere family even more than truth in legal proceedings. Two evidentiary rules, the spousal testimonial privilege and confidential marital communication privilege, were born. And they have for the most part survived.
The spousal testimonial privilege provides that spouses who are married at the time of trial have the right not to testify against each other. In most jurisdictions it's up to the spouse who would testify to claim the privilege. A few give the power to the spouse who's in hot water--in those states the spouse on trial can prevent the other spouse from testifying. Here's an example:
Kimmy and Tosha were unmarried partners living in a state that acknowledges gay marriage. The FBI arrested Kimmy for armed robbery. Tosha was with Kimmy immediately before and after the robbery in question and knew of Kimmy's plans to stick up a local bank. While Kimmy was awaiting trial, the couple got married. At Kimmy's eventual trial Tosha can legally refuse to testify. (In just a few states Kimmy would be able to prevent Tosha from testifying even if Tosha wanted to be a witness.)
The confidential marital communication privilege applies to personal conversations between spouses in private settings, even if the couple separates or divorces before trial. So, in the above scenario, let's assume Kimmy and Tosha were married before the armed robbery. When Kimmy returned home on the eventful night, she told Tosha, "That was a close call! I almost got shot!" In most jurisdictions, even if Tosha chose to testify against Kimmy at trial, Kimmy could prevent her from testifying regarding the statement about nearly being shot.
The DOMA decision doesn't seem to change anything about spousal privileges for same-sex couples. Individual states will apply the privileges according to their respective marriage laws: States that legalized gay marriage will afford them to same-sex spouses, while those that didn't won't. Federal courts apply the privilege rules of the states in which they are located. They are therefore likely to provide the spousal privileges only to those same-sex pairs who are legally married under state law.
Application of the spousal privileges is just one way in which Windsor shows us how much work is left to achieve equality. The decision didn't say that gay and lesbian couples have a right to marry. All it said was that the federal government must acknowledge same-sex marriages and provide rights and responsibilities accordingly. That means that state and federal courts in regions of the country that still "defend" traditional marriage will not offer the spousal privileges to same-sex couples who aren't legally married.
Whether or not spousal privileges have a place in modern society, applying them on the basis of sexual orientation doesn't. Let's hope that the next step in social equality is recognizing a fundamental right to marriage for everyone.