First a refresher on the definition of a "trade balance". This is essentially net exports minus net imports. Exports being when we, for example, create a watch in the USA and ship to Europe for sale; imports being when we purchase a european watch for sale in the USA. Oil is one of the many items in the import/export calculation.
A trade deficit is what occurs when we import more than we export (ie exports minus imports = negative number). When we run a trade deficit, we are, from the standpoint of trade, devaluing our currency. This is because we are putting more dollars on the foreign exchange market (imports; pay OUT dollars) than we are demanding (exports; COLLECT dollars).
When our currency is devalued, a number of things happen. First, all things being equal, it costs more to purchase goods from foreign suppliers, because our dollars buy less. This in turn increases the import bill, assuming we continue to purchase the same quantity of items from foreign countries. This would then become a downward spiral in our currency if it werent for a number of other factors, including the fact that we would NOT continue to purchase the same quantity of foreign goods.. as those goods became more expensive, we'd slow our purchases either out of necessity, or perhaps because now a domestic company can produce these goods more affordably. As you may see, supply and demand work together on many levels to balance these forces over the long term.
There are other things as well that impact the value of the dollar, so it's not all about imports/exports. Other factors contributing to the value of the dollar are prospects for interest rates and the general economic environment. All of that said, when you are in a situation similar to where we have been, in that interest rates have been decreasing, inflation increasing, large trade deficit, as well as an uncertain/negative outlook for economic activity, you have a major problem for a currency, in this case the US dollar.
When we look at oil and our dependence on foreign countries for this commodity, it's not a pretty picture. If the price of oil continues to rise, as it very well may, and we do not either curb our consumption (note that a recession will definitely put a damper on the consumption) or begin to produce more oil domestically, whether through increased exploration and development or through opening reserves (which I am opposed to. . .another story), then we will continue to see exponentially higher volumes of US dollars being sent elsewhere to pay for oil. This, for the reasons stated above, is not a pretty prospect for the US dollar. In addition, if the dollar continues to lose value (especially vs. our biggest import partners), then there will be upward inflationary pressures (which also hurt the dollar - see "Real Interest Rates"). Combine that with this hypothesized increased cost of oil and you're looking at other inflationary pressures to boot, such as higher input costs at farms (ie food prices) and manufacturers (ie prices of a variety of items).
Oil is a big concern and an important issue with regard to the future of the US economy and trade.