Using a homemade 776.9 nm semiconductor laser as seeder, a 2.3 W fundamental frequency output is obtained through amplification of a tapered amplifier (TA) and shaping. Then, a 388.4 nm continuous-wave laser is generated by the second harmonic generation (SHG) of a Class I phase matched lithium borate (LBO) crystal with an external bow-tie cavity. It is shown that the severely thermal effect of TA can change the waist position of the generated laser beam and limit the conversion efficiency of the second harmonic generator. Therefore, the thermal effect of TA is analyzed, and a laser beam shaping scheme consisted of cylindrical lens, prism pair and lens is proposed to compensate for the thermal effect. Furthermore, by optimizing the beam waist size and cavity parameters of the cavity, the conversion efficiency of the second harmonic is improved. Finally, the basic frequency input of 2.3 W can generate a ultraviolet laser output of 940 mW, with SHG efficiency of 41%.