Like all things, the perceived importance of an object is dependent upon which chair one occupies. In other words, a glass of water is pretty important if you're thirsty, but not so when there's a knife to your throat.
Obviously, I'm a bit biased on the issue of how important Bangladesh is to any international actor (whether state, non-state, or supra-state organization). For now (i.e. this post) I will limit my discussion to the importance of the Bangladesh to the relative position of the United States.
In some ways, this discussion should have occurred (i.e. I should have written it up) ages ago. But I assumed that if you're reading this blog then you're already aware of the geopolitical importance of the region. But I shall be clearer now: this region is important to the U.S.
Let's begin with the (perhaps unexpected) topic of trade. In 2011, over 4.8 billion U.S. dollars worth of imports arrived from Bangladeshi producers much of it probably finished textiles (check the collar of your shirt, go ahead, I'll wait). In terms of U.S. exports to Bangladesh this trade equaled over 1.1 billion U.S. dollars. Now before my American readers enter a protectionist fury over these numbers, let me urge you to reread some principles of macroeconomics textbooks. To sum, international trade has the net effect of lower prices in the destination market (i.e. the U.S.) on those goods, say ready-made garments. Now, this would be bad for RMG industrialists located in the U.S. but most are already located overseas and exporting the same (if not higher) quality goods for consumption back home at cheaper prices. It would be worth pointing out that current political campaigns don't even bother explaining this, because no one cares. And why not, clearly at some point Americans made their own clothes, but that was decades ago. This change has been absorbed.
This near 6 billion U.S. dollar trade ballooned upwards in 2011 from about 300 million in the 1980s to just about 1 billion in the 1990s. A big reason for this was the increasing liberalization of the Bangladeshi economy (particularly after Zia's coup in the 1980s). Comparatively, this trade volume is about medium its certainly smaller than China, India, or the United Kingdom (not to mention Canada and Mexico) but it is larger than say, Portugal (about 4 billion), Qatar (about 4 billion), the Czech Republic (about 5 billion), and is somewhat smaller than New Zealand (about 6 billion).
The strength of the trade relationship between the U.S. and Bangladesh should translate into sensitivity of the U.S. market to developments in the exporting sectors in Bangladesh (e.g. the RMG industry). The Bangladeshi press recognizes the importance of the ready-made garment industry to the country's economy (if not the export relationship outright) and can be counted on to run stories of RMG workers striking, protesting, and rioting over better working conditions, increased wages, and other benefits (of which there are few, hence why the country is attractive to textile firms in the first place).
So far, Bangla Nation has not covered any of the unrest in Bangladesh's RMG sector. But, to summarize this post, without the Bangladeshi government compromising with RMG workers to increase benefits, and perhaps bearing some of their cost while passing others to the corporations, the most important sector in the economy is at-risk. While the U.S. cannot intervene directly in this internal discussion, it can provide developmental aid to assist the Bangladeshi economy in diversifying to not only promote other sectors for export but to also incrementally raise living standards in the country opening additional markets for high quality U.S. goods (as an alternative to cheap Chinese goods).
Source Note: U.S. Department of Commerce, Bureau of the Census, Foreign Trade Division