In economic crises (of the Great Depression scale) money (especially cash) usually gains in value since people start liquidating other assets which become cheaper or just worthless. Since Bitcoin can be considered as both a currency and a financial asset, the overall effect is hard to predict. On the one hand, some people will have to sell their Bitcoin stashes to get the money with which they can buy food and pay the bills (for example, if they start losing jobs due to a crushing economic crisis). On the other hand, some other people, which are more wealthy and better off overall, may decide to rebalance their portfolios in favor of Bitcoin...
Since it is almost a given that governments will start printing an endless amount of money in an effort to arrest (or at least to alleviate) the consequences of the crisis
It will come down to what method the authorities decide to use to 'solve' the problem. In the periphery of the world, e.g. Greece recently, Asia in the 90s, and Latin America pretty much all the time, the global elites tend to force pure austerity onto the population. During the Great Depression in the US (the top of the world system,) a combination of deflation, stimulus, and devaluation against gold was used. During the 2008 crisis, the tools were almost exclusively stimulus and (de facto) devaluation against gold. The bottom line is that people are kept in more comfort nearer the top of the system, since the elites can't afford to generate too much anger among democratic populations.
The big deflationary effect on financial assets tends to be limited to real-economy assets, e.g. equities, real estate, industrial commodities, etc. because of the expected lack of economic demand. Money-class assets like cash and 'safe' government bonds that are issued by the state may be torn between the two forces of appreciation (by hoarding -- saving money instead of consuming/investing) and devaluation (by inflationary government stimulus.)
Money-class assets that are not issued by the state will be sitting pretty -- I think saver demand will outpace liquidation to buy food, etc., since there will still be a lot of financial assets of all types out there that people want to convert into these assets (remember what started the whole crisis is an overflowing quantity of all types of financial assets, typically concentrated among a minority of rich people.)