Having been granted a place in the top flight because of the collapse of Rangers, the Dens Park outfit have struggled and are nine points behind second-bottom Ross County following Wednesday's 1-0 loss to St Johnstone.
Some Dundee fans called for Smith's removal during that match and it had been suggested that he had lost the support of some members of the board.
However, the directors have decided to stick with Smith and are keen to stress the difficult assignment he and his staff were given.
Their statement read: "The board would like to thank the loyal Dundee supporters for their fantastic backing of the club this season.
"This is a season that was always going to be difficult, considering the extremely unusual circumstances that led to us being elevated into the SPL only one week before the League started.
"One day before the League started, we were officially admitted into the SPL. We then only had something like three weeks of the transfer window left to try and add to our squad and bring it up to SPL standard both in numbers and quality. A difficult job, it is fair to say.
"While the season has obviously not gone as well as we would have hoped for on the field and our injury list has been horrendous, there is no magic wand that can be waved to fix everything.
"We expect that the manager and the team will continue working hard in training and on match days to climb up the League but, in all reality, we were always going to find it tough against the established SPL teams.
"As a Board, we will redouble our efforts with the manager to try and add more quality to the squad and both the CEO and the manager have been working on targets to bring into Dens for the second half of the season.
"Media speculation over the last few days has caused great uncertainty at the club and this has been very unfair to the manager, the fans and to everyone involved at Dundee FC.
"The Manager, the CEO and the Board of Directors are confident we can still put on a good show in this division and that we can be in the mix to the very end. Once again, it is time for everyone at Dens Park to stand up and be counted and we hope you can stand with us."