A pair of hotly contested races in Nassau County on Long Island could help decide which party controls the New York state Senate, where Republicans wield control even though the Democrats hold a razor-thin majority.

One battle involves Democratic state Sen. Todd Kaminsky, who again faces Republican attorney Chris McGrath in the state’s Ninth Senate District after beating him in an April special election. That seat had been held for three decades by former Republican Senate Majority Leader Dean Skelos, who was convicted in...