At the time of the Amendment's passage, tariffs on imported goods and excise taxes on produced goods were the primary sources of government revenue. The burden of the tariffs fell disproportionately on the rural areas of the country (a much larger percentage of the population than today), since it raised the prices of the goods that they bought. They favored an income tax because it would largely be paid by urban workers and industrialist who had regular income, rather than them.

The industrialized Northeast opposed the 16th amendment for this reason and favored keeping tariffs high. Support for the 16th was weakest there, and strongest in the agricultural South and Mid-West.

First off it is important to keep in mind what the taxation powers of the federal government where held to be from 1895 before the passage of the 16th Amendment in 1913. The Constitution authorizes the government to collect indirect taxes such as excises and tariffs. However, the Constitution only authorizes direct taxes such as head taxes or property taxes if the burden is apportioned by states according to the population. Strange as it may seem, a tax on income earned from wages was considered an indirect tax (specifically an excise tax) and thus was within the power of the federal government to levy on individuals. Taxes on income earned from property in the form of dividends, interest, or rents (leasing fees) were considered after a Supreme Court case in 1895 to be direct taxes and thus could only be levied on the states in proportion to their population. As it happens, an income tax apportioned among the states was never implemented. The 16th Amendment was therefore not a huge increase in taxing power, it just allowed the federal government to tax income earned from property in the form of dividends, interest, or rents without apportioning the burden to the states and so was seen as leveling the playing field with taxes on wages.

TL;DR: At the time, the federal government could already tax wages similar to how it can now, the 16th Amendment just eliminated the privileged position of income derived from property

Sources: Ray Raphael's Constitutional Myths: What We Get Wrong and How To Get It Right https://en.wikipedia.org/wiki/Pollock_v.Farmers%27_Loan%26_Trust_Co.

EDIT: as /u/tim_mcdaniel pointed out below I originally mislabeled the tax on income derived from property as an indirect tax.