In a situation where more transactions take place online there will be fewer uses for having cash on hand. As more commerce and banking goes online, then the demand will rise for alternate payment systems like PayPal and Amazon Payments. There is also the rise of the crypto currencies that operate outside of the state-sponsored monetary system, such as bitcoin.
That said, electronic commerce is growing, but from a relatively low base. The Department of Commerce has been tracking the rise in e-commerce over the past twenty years. In the most recent quarter, online sales only accounted for 6.1% of total sales. The trend is clear though, more and more sales are happening on line, as illustrated in figure one.
The linear trend show that if it holds, it will still take over thirty-five years for all commerce to happen on line. Of course, that is if the linear trend holds. It has doubled in the last five years, so if it follows the trend where it doubles every five year, then it will be much faster.
If banking takes off as quickly as commerce did, there will soon be a reduced demand for holding cash – though there are still cultural biases to holding physical cash. What it will not do is decrease the supply money much. Currency in circulation is just a small part of the money supply (Krugman & Wells p. 425). Banked dollars that are accessible through checking are just as good as those that are folded in the wallet.
Krugman, P. & Wells, R. (2013). Macroeconomics. New York, NY : Worth Publishers
Thomas, I., Liu, X., & Weidenhamer, D. (2014, November 18) Quarterly Retail E-Commerce Sales: 3rd Quarter 2014. U. S. Census Bureau News. Washington, D. C. Retrieved at http://www.census.gov/retail/mrts/www/data/pdf/ec_current.pdf