The most common use of a solution of BSA dissolved in water would be as a series of BSA solutions of different concentrations for Bradford assay or total protein assay.
For applications other than protein quantification, BSA is normally dissolved in buffers such as PBS, TBS, PBS-T, and TBS-T. At a concentration of 1-2%, BSA in PBS/TBS may be used as a blocking solution in ELISA reactions, whilst at a concentration of ~5%, BSA in PBS-T/TBS-T functions well as a blocking solution in Western Blot reactions. BSA is always preferred in Western Blot reactions which involve analysis of phosphorylated versions of proteins, as the alternative, skim milk, might contain phospho-casein which would yield a background with high noise.
Note: BSA might foam substantially when dissolved in any solvent; thus, one must wet the BSA powder first and gradually add the solvent to properly dissolve the powder. Some might even suggest to just leave the BSA powder to dissolve without shaking at r.t.p. for 10 min as it would dissolve quite easily in due time.
It really depends on what you're using it for. If you want to measure protein concentration using a series of BSA concentrations as standard, you'd want the BSA solutions to have the same additives that your protein solution have. For Bradford you'd avoid detergents, for BCA you'd likely avoid mercaptoethanol, so it's very procedure dependant. For general storage a basic buffer tends to be used given that the isoelectronic point of BSA is 4.7, so you want to avoid acidification by carbon dioxide in more concentrated solutions to avoid precipitation. I think the most likely answer is use the same medium if you're using it to measure concentrations. If you're using it to block, use it as is since no significant chemical changes happen.