There are four ways I can think of; the rest of you are welcome to chime in if you know any others.
The first way to financially independent transit is simply the farebox. If you can get passengers to pay for all your operating and maintenance costs, and maybe even some of your capital expansion costs, you're golden. This is why borrowing is a bad idea, though: the more farebox revenue you spend on debt service, the more ongoing subsidies you'll need to ask for. Other challenges include losing passengers to competing modes, and being politically or economically unable to raise fares.
As we saw in the news articles that started this whole discussion, the vast majority of public transit agencies in the US are unable to fund their operating costs - in many cases, even when ridership is high - and depend on subsidies out of general tax revenue.
The second way is to run a side business. The MTA brings in money from advertising and from renting space on its properties for newsstands and other businesses. Ben Kabak of Second Avenue Sagas is a big fan of this revenue source, but it's important to place some value on transit riders' dignity and sanity, and I personally can't stand being bombarded with ads all the time. It also compromises the system's integrity if the managers are chasing ad revenue.
Another strategy is to tax a specific thing and dedicate the revenues to transit; the most salient example of this is the Triboro Bridge and Tunnel Authority, whose bridge revenue goes to fund the New York City Transit Authority. The MTA also gets revenue from the Mortgage Recording Tax, but that's a double-edged sword. First, with hardly anyone taking out mortgages these days, revenues from that are way down. Second, there have been a large number of real estate speculators involved in MTA management recently, and I have to speculate that there was some kind of condition on the continued collection of the mortgage recording tax where they were allowed to have some say in how its proceeds were spent. Since most of the powerful real estate people don't take transit (former MTA chair Peter Kalikow owns a large collection of Ferraris), it's safe to say that they may not always have the best interest of the transit riding public at heart.
The fourth way to independence relates to the first way, and that's to bring down operating costs through investment. This investment can be subsidized by the government, but it often doesn't have to compete quite as fiercely for funds as operating revenues do. The type of investment that contributes the most to financial independence is dedicated, separated right-of-way. This is why the Lincoln Tunnel Exclusive Bus Lane accounts for so many of the profitable private transit operators in the country. It reduces the travel times for the routes that use the tunnel, and thus saves gas and operator wages. There are other ways: before the "private" bus lines in New York City were taken over by the MTA, most of their buses were purchased and owned by the City.
Recently, many people have wondered why the Federal government subsidizes many capital costs of transit systems, but is reluctant to subsidize operating costs. The main reason is that it's a lot easier politically. Give a man a fish and you've fed him for a day. Give him a fishing pole and he probably won't ask for another one for at least a month.
So what can these cash-strapped transit agencies do? One thing is to identify capital improvements that can lead to significantly lower ongoing costs. New buses, garages, signal priority, anything that facilitates operations. Of course the things that facilitate bus operations the most are busways. And the thing that facilitates transit operations the most is rail.