Ohio Rev. Code Ann. § 3513.19
(A) It is the duty of any precinct election official, whenever any such official doubts that a person attempting to vote at a primary election is legally entitled to vote at that election, to challenge the right of that person to vote. The right of a person to vote at a primary election may be challenged upon the following grounds:
(3) That the person is not affiliated with or is not a member of the political party whose ballot the person desires to vote. Such party affiliation shall be determined by examining the elector's voting record for the current year and the immediately preceding two calendar years as shown on the voter's registration card, using the standards of affiliation specified in the seventh paragraph of section 3513.05 of the Revised Code. Division (A)(3) of this section and the seventh paragraph of section 3513.05 of the Revised Code do not prohibit a person who holds an elective office for which candidates are nominated at a party primary election from doing any of the following: