How Do Economists Quantify Contagion?
I believe the single most commonly used metric for contagion is a sharp increase in the correlation of returns in various countries, either within a country (e.g., the correlation of daily returns in the markets for stocks, bonds, and commodities) or across countries (e.g, the correlation of daily returns in the stock markets in Europe, North and South America, Asia, etc). 

While correlation is an easily understood measure, and while it's true that pairwise correlations do tend to rise during crisis periods, using correlations in this manner is not without pitfalls. In particular, sampling correlation coefficients will rise during periods of heightened market volatility (and this is a major difference to, say, regression coefficients), irrespective of whether there's a real crisis or not. For more on these pitfalls see, e.g.,

Forbes and Rigobon, 2002, "No Contagion, Only Interdependence: Measuring Stock Market Comovements", Journal of Finance, vol. 57, pp. 2223-2281.

and

Loretan and English, Evaluating changes in correlations during periods of high market volatility, BIS Quarterly Review, 2000, June, pp. 29-36.