You can use rgb() to specify color so you can control alpha:
plot(1:5, cex = 0)
points(1:5, pch=20, cex=2, col = rgb(0, 1, 0, c(0.1,0.2, 0.3, 0.4, 0.5)))
In your example:
points(DF$x_values, DF$y_values, pch=20, cex=2, col = rgb(0, 1, 0, alpha))