Have straight party-line votes (or nearly so) become more common in the US Congress? If so, how long has this trend been happening?

As time has progressed, partisanship (especially in the House) has increased drastically. This can be visualized here. During the New Deal party system (1932-roughly 1968), there were many Democratic office holders that ideologically identified as conservatives, and many Republican office holders who identified as moderate or liberal. Examples of these "Rockefeller Republicans" include Sen. Bob Packwood, Wayne Morse, and of course the namesake Nelson Rockefeller. There were many conservative Democrats (primarily from the south) as well.

Therefore, votes on the Senate floor in the 1960s may have appeared "bipartisan" by today's standards, but will break down if you apply any ideological scrutiny. Let's take a look at a piece of legislation from this time period, the Civil Rights Act of 1964. First, look at the roll call here.

As you can see, the bill passed with the support of 46 Democratic votes, and 27 Republican votes. The "neas" consisted both of Democrats and Republicans as well. Like we talked about previously, the southern Democrats voted almost unanimously in opposition along with a small group of conservative Republicans like Sen. Barry Goldwater of Arizona. Had we organized these legislators by their ideological identification we might have seen a vote that looked like 73 yea (73 liberal/moderate) and 27 nea (conservative).

The "drop in bipartisanship" can be attributed to this ideological realignment that began after the 1968 presidential election, and arguably reached fruition by the 1994 midterm elections. Other factors include the abolishment of the Fairness Doctrine, the reduction in the number of "bridge-builders" in Congress, gerrymandering, and closed primaries. All of these contribute to the modern-day hyper-polarization and gridlock experienced in Washington.