The title of this diary may seem intuitive. Liberal areas elect liberals. Liberals like public transit, both for economic and environmental reasons. Conservative areas don't elect liberals. Conservatives like public transit less, both for economic and environmental reasons. However, I decided to actually see how strong the correlation is, and the results are so incredibly strong I wanted to share them anyway.
First, I want to briefly explain my methodology.
Urban Area = The Census-defined urban areas. However, I combined San Francisco and San Jose because, as a local, I can tell you it is one continuous urban area. Otherwise, I stuck precisely to the Census definitions.
Democratic Index = I used the Cook PVI of various urban areas, which I mapped out on Dave's Redistricting App as best I could. This requires using 2008 numbers, which are a bit out of date. I made adjustments for Chicago (-2%), Phoenix (+3%), Tucson (+3%), and Indianapolis (+3%), due to the weird 2008 results in all four areas due to candidate and campaign effects. I considered doing the same for Kansas City, St. Louis, and New Orleans, all of which had somewhat weird 2008 results, but decided there wasn't enough evidence. For example, the New York urban area voted 68% for Obama in 2008. Since Obama got about 54% of the national vote, its Democratic index is 64. In an neutral election, that's what the Democrat would get.
Public Transit Index = I used the American Public Transportation Association ridership report, which gives 2013 numbers for almost every urban area with 750,000 or more people. I used the metric of rides per person per year. I simply divided the number of unique rides of public transit by the number of people in the area to obtain a number. Pretty simple. One thing I will note (that I believe especially affects New York and DC) is that tourists can cause artificially high numbers. However, I see no way to adequately adjust for this, so I left it be.
I then graphed an easy Excel scatterplot and saw that the numbers lined up closely. Very closely. In politics, anything above an r squared of 0.25 is pretty great. (For those who don't know stats or need a refresher, 0.25 means that the two factors are linked with one another in a positive direction and that 25% of the variation can be explained by this link)
When I graphed it, I immediately noticed two outliers: New York and Salt Lake City. Even with this, however, there was an r squared of 0.31. 
That's pretty high. As you can see, New York's ridership far, far outpaces anything else. It's the one on the far right. Salt Lake City is the one on the very bottom; it is by far the most conservative urban area in the country over 750,000 people (although Oklahoma City, another very red one, isn't on this chart because there wasn't data). Eliminating these two outliers, both of which I believe are unique cases (NYC has tons of tourists that use the subway and is far denser than anywhere else, and Salt Lake City has thousands of people who would be Democrats if not for their religion to an extent not true anywhere else in the country), the graph looks like this:
Now that's some serious correlation. So, in summary, if you want to know how liberal or conservative an area is and know nothing about politics, just look at its public transportation ridership.
What's Missing: Unfortunately, the APTA excludes 6 populous urban areas from its results: Las Vegas, Tidewater, Oklahoma City, Richmond, Jacksonville, and the Research Triangle. DRA also doesn't have results for Portland, Providence, or Honolulu. Therefore, only 43 of the 52 areas I wanted to use are here. I still think the results are strong.
Briefly, here are the 5 outliers:
New York City: As I said, it's just different than anywhere else in the country. Denser, the most tourists, incredible subway system.
Salt Lake City: I hadn't though about this until now, but I think Utah would be just like Colorado politically if it weren't Mormon. Salt Lake City is already as blue as Denver, but the suburbs would be swingy like Denver's if they weren't full of Mormons. St. George would still be red, like Colorado Springs, and the northern Wasatch Front would probably be purple or light red like Fort Collins/Greeley. Everywhere else essentially has too few people to matter. The people in Utah are no different from those in Colorado except for their religion.
Detroit: Detroit has a very low public transit rate. Of the 43, it comes in 41st. However, it's a pretty blue area. My guess is that the racial divide in the area stopped construction of commuter rail during the 1960s or 1970s, and the lack of money in Detroit means they can't fund an adequate bus system.
Hartford: Hartford is far bluer than it should be, demographically speaking. It's probably due to being in New England, although Boston wasn't an outlier.
El Paso: El Paso is in Texas, which lags in public transportation across the board (San Antonio is 21st, Austin is 22nd, El Paso is 25th, Houston is 30th, and the Metroplex is 33rd). The racial demographics are also different; it's the only majority-Hispanic urban area over 750,000 people.
A Couple Fun Lists
Top 10, with scores: New York 220, DC 100, Bay Area 97, Boston 94, Chicago 74, Seattle 66, Philadelphia 65, Los Angeles (surprisingly high given all the crap they get for having no public transit) 51, Baltimore 48, Salt Lake City 43. These tend to be among the 10 largest, with Seattle, Baltimore, and Salt Lake City all better than they should be and the Sun Belt areas all off the list when they should be on it.
Bottom 10, with scores: Birmingham 4, Indianapolis 7, Detroit 8, Memphis 10, Kansas City 11, Nashville 11, Tampa Bay 12, Cincinnati 13, Inland Empire 13, Columbus 14.
Abnormally Low, Given Population: Miami, Metroplex, Houston, Detroit, Phoenix, Tampa Bay, Inland Empire, Indianapolis, Birmingham
Abnormally High, Given Population: Bay Area, DC, Seattle, Boston, Portland, Milwaukee, Salt Lake City, Buffalo, Hartford, Honolulu