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 appears faded and is unclickable, it indicates that the Campaign has been successfully activated and the messages are currently in the queue for delivery. Once all queued messages have been dispatched, the Edit button will become active again.
If you activated the campaign more than an hour ago -
If you activated the campaign over an hour ago, it is likely that most or all of the messages have been sent, depending on how many were sent at once.
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 select the Track button to view the number of messages sent and the specific delivery times for each.