There are a couple of different ways you can confirm if a campaign has been sent and the method you should use will depend on how long ago you activated it.
If you just activated the campaign a couple of minutes ago -
First you will want to return to the main campaigns screen for the campaign you activated, in this example we’ll say a Text Message campaign.
Go to Campaigns and then select Text Message

Scroll down to where the Campaign you selected to send is listed.

If the Edit button next to the Campaign is faded and cannot be clicked then that means that the Campaign has been activated and the messages are queued for delivery. After the queued messages have been sent the Edit button will return to the normal function.
If you activated the campaign more than an hour ago -
If you activated the campaign more than an hour ago then all or most of the messages will be sent depending on how many were sent out at one time.
To confirm that it has gone out you will follow the same steps as above but you will not be looking for the Edit button to be faded. Instead, you will see a date for Last Sent in green next to the campaign like this -

You can also click on the Track button to see how many went out and what times they were delivered.