Do modern economists still believe balance of trade is an important concept in economics? My understanding is that discussions about a nation's balance of trade was largely seen as important by mercantilists, but isn't really part of modern economics. Am I right about that?
The economics of trade can broadly be divided into two parts.  There is a microeconomic part concerned with patterns of trade in particular kinds of goods, their causes such as factor endowments and costs, and effects on welfare, key concepts being comparative advantage and the gains from trade.  Then there is a macroeconomic part concerned with overall volumes of trade and payments for traded goods, and their relations with exchange rates and domestic macroeconomic conditions.  The two parts can be linked, as in the Dekle et al paper referred to by emeryville, but more often are considered separately.

The concept of balance of trade belongs in the macroeconomic part, although 
economists often use the more precise term balance on current account, which includes trade in goods and services and certain other items. It is generally considered that a continuing deficit on current account is harmful because the net outflow of funds cannot indefinitely be met by running down the country's international currency reserves, and must eventually be balanced by an inflow of funds associated with increasing international debt and/or foreign ownership of domestic assets.  The moderate aim of a balance on current account avoids two disadvantages of mercantilism, which aimed to achieve a surplus: firstly, it led to conflict because it was impossible for every country to have a surplus, and secondly, it reduced welfare, because restrictions on imports meant that the (microeconomic) gains from trade were not fully achieved.

Thus the balance of trade or balance on current account remains an important concept in the macroeconomics of trade.