Unfortunately there is no read receipt feature, although it is on their dev backlog according to this Tweet thread.
There isn't much you can do outside of asking if they saw it. Some other possibilities:
Look at their screen to see if their Slack app is showing the unread message notification. This isn't perfect though. You'll know they read it if the notification isn't there, but they may have also seen it even with the notification showing, due to other unread messages.
If they are offline when you message them, wait around to see if they appear online shortly after you send the message. That is a likely indication that they read it. Having said that, it might be possible that a user's account is "sleeping" and the act of receiving the message sets them as online. You can test this with someone that has the Slack app on their phone. Have them login and leave the app alone for some time. Once they no longer appear online, send them a message and see of they come online without having to reopen the app.
Open a direct message and STAR it to keep it showing on the sidebar. This is very useful because it does remove any duplicates that would show up in the direct messages part of the sidebar, so you still have plenty of space.
Best Answer
Click on "Direct Messages" to bring up a list of recent conversations: