The NYT seems very concerned that the dollar will fall if the budget deficit is not reduced. Usually economists believe that a large budget deficit will increase the value of the dollar. The logic is that higher budget deficits are believed to cause higher interest rates, which makes holding bonds and other dollar denominated assets more attractive. This is how a budget deficit can cause a trade deficit.
The mechanics of this process are somewhat dubious in that there is very little relationship between budget deficits and trade deficits. (In 2000, when the country was running a huge budget surplus, we also had a large and rapidly growing trade deficit.) However, there is a relevant accounting identity which is always true. The trade surplus is equal net national savings. This means that if we have a trade deficit, then net national savings must be negative. The implication of a large trade deficit is that either public savings must be very low or negative (i.e. a large budget deficit) and/or we must have very low private savings. There is no possible way around this accounting identity.
This means that if the U.S. has a large trade deficit, as it currently does, then it must be the case that either households have very low saving or the country has a budget deficit. At the peak of the housing bubble, private saving was very low, since households spent based on their housing bubble wealth. Now that much of this bubble wealth has disappeared with the collapse of house prices, saving has moved back toward more normal levels. This means that to sustain the same level of output, the budget deficit must rise. There is no way around this identity.
A drop in the value of the dollar is the main mechanism for adjusting the trade deficit. This decline is exactly what would be expected in a system of floating exchange rates. However, the people who are concerned about the decline in the dollar, and also want the U.S. government to reduce its budget deficit, must want to see the level of output in the United States to fall and its unemployment rate to rise. That is the only plausible way that the accounting identities can be kept in balance.
The NYT should have pointed out to readers that the people concerned about the decline in the value of the budget deficit lowering the value of the dollar apparently want to see an increase in the unemployment rate in the United States.
This article also includes inappropriate adjectives, like "huge" before "budget deficit" and "expensive" before "entitlement programs." Such adjectives should be left to the opinion page. This would be a more accurate and shorter article without them.