There isn't a village shop in Bassenthwaite, but there is a seasonal shop on the Trafford's Caravan Park at Bassenthwaite, which is just on the A591 heading towards Carlisle. The shop is open seasonally, from April through to October. The next nearest shops and supermarkets are found in Keswick and Cockermouth.
Mireside Farmhouse is roughly equidistant from the main towns of Keswick and Cockermouth. The latter is a pleasant market town with a good range of shops and several small supermarkets including Sainsburys and Aldi. It also has less traffic congestion in the busy season.
Keswick has a traditional market in the main square on Thursdays and Saturdays, Booths is Keswick's largest supermarket and has a wide variety of ready meals (frozen & fresh), as well as an extensive wine and beer department. There is also a Co-Op in Keswick just a short walk from the main square. Keswick also has a wide selection of outdoor clothes and equipment shops including George Fishers, Cotswold Outdoor, Blacks, Millets, and Rathbones.
For more, mainstream shopping you will need to take a trip to Carlisle or Workington, where you will find nationwide stores such as Debenhams, Marks & Spencers, House of Fraser, etc.