As others have said, it's a perfectly safe an acceptable practice to fire .38 Special, .38 Special +P, and even .38 Special +P+ ammunition in a .357 Magnum revolver.
.38 Special is the parent cartridge for the .357 Magnum and the Magnum itself was an outgrowth of high-pressure .38 Special loadings in the 1920's and 1930's. Originally, S&W produced two revolvers on their large N-Frame for a cartridge they dubbed the .38-44. The .38-44 was dimensionally identical to a standard .38 Special cartridge, but loaded to much higher pressure and velocity (it would be considered a +P+ loading today). While the large S&W N-Frames and other comparatively strong revolvers like the Colt Single Action Army and New Service could handle .38-44 ammunition just fine, the same ammo would also chamber and fire in older, smaller, weaker guns like S&W K-Frames and Colt D-Frames that couldn't take the pressure (K-Frames were later chambered in .357 Magnum, but that was in the 1950's when different heat-treating was available).
Because of this, S&W and Winchester got together and, with input from both Phil Sharpe and Elmer Kieth, came up with the .357 Magnum cartridge. The case was lengthened to prevent it from chambering in .38 Special revolvers and it was found that the extra case capacity allowed the cartridge to be loaded to levels beyond what even the .38-44 cartridge could achive. Original advertised ballistics were a 158gr bullet at 1500fps from an 8 3/4" barrel and, from chronograph tests I've seen on vintage ammo, those estimates weren't all that far off. As I recall, the .357 Magnum held the title of "most powerful handgun in the world" until the .44 Magnum debuted in 1956.
SAAMI maximum pressure for the .357 Magnum cartridge is currently 35,000 psi (this has been reduced somewhat as I've seen data to indicate that the cartridge was originally loaded to over 40,000 psi) while .38 Special max is 17,000 psi and .38 Special +P max is 20,000 psi. Because the outside diameters of the case and bullet diameter of both .357 Magnum and .38 Special are identical and because, unlike semi-auto which headspace on the case mouth, most if not all .357 Magnum revolver headspace on the cartridge rim there is absolutely no safety issue with shooting .38 Special ammo in a .357 Magnum revolver.
The only issue which is commonly encountered as a result of the practice is that the shorter .38 Special cases will leave a ring of fouling farther back in the chambers than .357 Magnum cases will. If .357 Magnum ammunition is fired in a revolver which has previously been fired with large amounts of .38 Special ammo and has not been thoroughly cleaned, chambering and extraction of the longer magnum cases will likely be sticky. If the problem is allowed to go on, the rings of foulind could possibly cause pressure spikes with magnum ammo, but that is only in extreme cases. The whole issue can be avoided by simply cleaning the revolver thoroughly after using .38 Special ammunition.
