Let me start from the end: what would happen to the U.S. economy if the trade deficit were to worsen? The answer is, it depends.
Normally, trade deficit is offset by capital account surplus; American households, businesses, and government buy more goods and services abroad than they sell abroad (this is current account deficit), but sell to foreigners more assets (stocks, bonds, land, and real estate) than they buy abroad (this is the capital account surplus). In case of the U.S., a large part of the current account deficit is offset by purchases of the U.S. Treasury bonds by Bank of Japan and Bank of China, as well as by acquisitions of U.S. companies by European companies.
Sometimes, foreigners don't have enough interest in the country's assets to fully offset the current account deficit, so situation adjusts itself through devaluation of the country's currency. Devaluation benefits exporting industries and those who invest abroad; it hurts importers and foreign investors. Net effects of devaluation are usually small (unless devaluation is fast and drastic, such as those that used to happen in Mexico in the years of presidential elections). Large devaluations of the dollar have occurred in the past; for example, between 1985 and 1987, the U.S. dollar lost half of its value compared to the Japanese yen and the German mark. Interestingly enough, this devaluation didn't cause a recession.
Now back to the beginning: why shouldn't a country have a current account deficit? This is a question that is often asked by people who wish to divert attention from a more important problem, that of the BUDGET deficit...