How about:
SELECT * FROM TRIPS LEFT JOIN CHECKINS
USING (MemberID) ORDER BY CheckInDateEntered ASC;
(http://sqlfiddle.com/#!2/7f6055/3)
You can also order by multiple fields separating them by commas, in order of priority:
SELECT * FROM TRIPS LEFT JOIN CHECKINS USING (MemberID)
ORDER BY VenueCreatedAt,TripDateStarted,TripDateEnded ASC;
(http://sqlfiddle.com/#!2/7f6055/10)
And if that doesn't work either, check out the LEAST function:
SELECT * FROM TRIPS LEFT JOIN CHECKINS USING (MemberID)
ORDER BY LEAST(VenueCreatedAt,TripDateStarted,TripDateEnded) ASC;
(http://sqlfiddle.com/#!2/7f6055/15)
But be aware this will only sort by the one column with the lowest date. In case you want to sort by the minimum value on each row (alternating columns), maybe any of these three other questions may help you:
Order by max value in three different columns
Select Smallest Value From Multiple Columns with PHP/MySQL
Sorting a MySQL query with ORDER BY or with PHP sort functions
(they are all about sorting by the minimum or maximum value on each row of a particular set of columns)