What legal recourse does a wife have against the woman who is having an affair with her husband?

You have little or no legal recourse against her. Time was, you could sue for "Alienation of Affection", but it now seems to be seen as a relic of a bygone era, and very few states still recognize it. Here's what WorldLaw Direct says about it:

http://www.worldlawdirect.com/article/1026/Alienation_of_affections_--_Law_suits.html

Alienation of affection(s) is a legal action, a tort based on willful and malicious interference with marriage relations by a third party. The elements constituting the cause of action are wrongful conduct of the defendant, plaintiff’s loss of affection or consortium of spouse, and a causal connection between the two. Not all states recognize the right to bring an alienation of affections action. It is usually viewed as not being relevant or easily applied in modern society.

States plus D.C. which have statutorily (i.e. by legislation) abolished alienation of affection:

Alabama
Arizona
Arkansas
California
Colorado
Connecticut
Delaware
District of Columbia
Florida
Georgia
Indiana
Kansas
Maine
Maryland
Massachusetts
Michigan
Minnesota
Montana
Nebraska
Nevada
New Jersey
New York
North Dakota
Oklahoma
Oregon
Pennsylvania
Rhode Island
Tennessee
Texas
Vermont
Virginia
West Virginia
Wisconsin
Wyoming

States which have judicially (i.e. by court ruling) abolished alienation of affection:

Idaho (1986)
Iowa (1981)
Kentucky (1992)
South Carolina (1992)
Washington (1980)

States in which alienation of affection is possibly a viable cause of action:

Hawaii
Illinois
Mississippi
Missouri (appears recently abolished)
New Hampshire
New Mexico
North Carolina
South Dakota
Utah

[To succeed on an alienation claim, the plaintiff often must show that (1) the marriage entailed love between the spouses in some degree; (2) the spousal love was alienated and destroyed; and (3) defendant’s malicious conduct contributed to or caused the loss of affection. It is often not necessary to show that the defendant set out to destroy the marital relationship, but only that he or she intentionally engaged in acts that likely would impact the marriage.]