When a stock splits does the dividend go down?

Yes and no. Here's an example to illustrate:

Let's say that you own 100 shares of XYZ Corp., which pays 50¢ per share in dividends. That means that you get $50 in dividends for your 100 shares. Now the stock splits 2-for-1. You now have 200 shares of XYZ Corp., but the dividend is now 25¢ per share. However, your total dividend for the amount of stock you own is still $50.