I pronounce it in the good ole American way - Julius Caesar. Not with a hard C. But like most Americans I tend to pronounce foreign words as they are said in the language of origin, although I tend not to use Latin pronunciation.
My favorite Brit-speak: the Italian film director Pasolini (Pass-O-Lini) is pronounced Pass-AHL-oni in Great Britain. Truly dreadful.
If you don’t pronounce it ‘Yoolius Kayser’ you are guilty of perpetuating one of those English mispronunciations you are railing against.
The only way I can think of that this mispronounciation could have come about is if some English person who had never heard Latin being spoken just read the name as he thought it looked and passed this error on to others. The change in pronounciation could only have come about as a result of confusing the hard and the soft uses of the written letters ‘j’ and ‘c’...