As far as I know, no states elect their state senators by county. Why not? US senators are elected by state, so wouldn't it make sense for state senators to be elected by county?
The US Senate was designed to entice the smaller states to surrender their sovereignty by ensuring that they would always have a relatively powerful voice in the national government. The various counties were never sovereign so there was no need to entice smaller counties to agree to join the state. Additionally, at the time that the various state constitutions were being written, there was no legal requirement that districts have equal population. If powerful interests wanted to ensure they remained powerful, they could simply draw small districts that they could control so there was no benefit to having a state senator from a small county that had the same power as a state senator from a large, populous county.
The Legal Information Institute at Cornell Law School has an online summary of the "one person, one-vote rule". Since counties have different numbers of people living in them, if each county got the same number of senators, some voters would get more powerful representation in the state senate than others.
One way to solve this would be to vote for senators on an at-large basis, and elect more senators from some counties than others. But there are lots of problems with at-large elections, where the N candidates who get the most votes are elected.
It used to be that some US states did exactly this. In a series of cases kinitially known as the "one man, one vote" cases (later "one person, one vote"). The US Supreme Court held that state legislative districts must be equal in size, as measured by population.
Reynolds v. Sims, 377 U.S. 533 (1964) dealt with the state of Alabama. In 1964 the Constitution of Alabama provided for one state senator from each county regardless of population differences.
According to the Wikipedia article on the case [citations omitted]:
Before the industrialization and urbanization of the United States, a State Senate was understood to represent rural counties, as a counterbalance to towns and cities. State and national legislatures had been reluctant to redistrict., because there existed general upper-class fear that if redistricting to meet population changes were carried out, voters in large, expanding or expanded urban areas would vote for confiscatory wealth redistribution that would severely inhibit the power of business interests who controlled state and city governments early in the century. Of the forty-eight states then in the Union, only seven twice redistricted even one chamber of their legislature following both the 1930 and the 1940 Censuses. Oregon did not redistrict between 1907 and 1960, Illinois not between 1910 and 1955, while Alabama and Tennessee had at the time of Reynolds not redistricted since 1901.
In the majority opnion the Court wrote:
Undeniably, the Constitution of the United States protects the right of all qualified citizens to vote, in state as well as in federal, elections. A consistent line of decisions by this Court in cases involving attempts to deny or restrict the right of suffrage has made this indelibly clear. It has been repeatedly recognized that all qualified voters have a constitutionally protected right to vote, Ex parte Yarbrough, 110 U. S. 651, and to have their votes counted, United States v. Mosley, 238 U. S. 383. ...
The right to vote freely for the candidate of one's choice is of the essence of a democratic society, and any restrictions on that right strike at the heart of representative government. And the right of suffrage can be denied by a debasement or dilution of the weight of a citizen's vote just as effectively as by wholly prohibiting the free exercise of the franchise.
In Baker v. Carr, 369 U. S. 186, we held that a claim asserted under the Equal Protection Clause challenging the constitutionality of a State's apportionment of seats in its legislature, on the ground that the right to vote of certain citizens was effectively impaired, since debased and diluted, in effect presented a justiciable controversy subject to adjudication by federal courts. The spate of similar cases filed and decided by lower courts since our decision in Baker amply shows that the problem of state legislative malapportionment is one that is perceived to exist in a large number of the States.
In Gray v. Sanders, 372 U. S. 368, we held that the Georgia county unit system, applicable in statewide primary elections, was unconstitutional, since it resulted in a dilution of the weight of the votes of certain Georgia voters merely because of where they resided. After indicating that the Fifteenth and Nineteenth Amendments prohibit a State from overweighting or diluting votes on the basis of race or sex, we stated:
How, then, can one person be given twice or ten times tvoting power of another person in a statewide election merely because he lives in a rural area or because he lives in the smallest rural county? Once the geographical unit for which a representative is to be chosen is designated, all who participate in the election are to have an equal vote -- whatever their race, whatever their sex, whatever their occupation, whatever their income and wherever their home may be in that geographical unit. This is required by the Equal Protection Clause of the Fourteenth Amendment. The concept of 'we the people' under the Constitution visualizes no preferred class of voters, but equality among those who meet the basic qualifications. The idea that every voter is equal to every other voter in his State, when he casts his ballot in favor of one of several competing candidates, underlies many of our decisions.
[I intend to add to this answer later.]