If I wanted to help a family member who is going through a difficult financial time and give monetary gifts, how would this be found out? For example, if I gave my sister $5,000 one month and then a few months later, let's say, $4,000 and a few months later, another check. Let's say, it totals more than $13,000 in one year. How does the IRS find out about this? Do banks report checks that go over a certain amount?

The IRS won't find out unless they do an audit of either one of you.
Banks report CASH, actual green cash, 10K and up.

But there are other ways to find out, such as lifestyle audits.

Understand that even though you give MORE than $13,000, you need only FILE a gift tax return.

Due to the Unified Credit, you will not actually PAY any gift taxes until you exhaust the credit, which does not happen until the git amount exceeds about $2 MILLION.

(14K this year, 2013)

Noted