We propose the broadband mode-selective coupler (MSC) formed with a side-polished six mode fiber (6MF) and a tapered side-polished small core single-mode fiber (SC-SMF) or an SMF. The MSCs are designed to allow the LP01 mode in the SC-SMF and SMF to completely couple to the LP01, LP11, LP21, LP02, LP31, LP12 modes in the 6MF over a broadband wavelength range. The phase-matching conditions of the MSCs are satisfied by tapering the SC-SMF and SMF to specific diameters. The tapered fibers are side-polished to designed residual fiber thickness using the wheel polishing technique. The effective indices of the side-polished fibers are measured with the prism coupling method. The MSCs provide high coupling ratio and high mode purity. High coupling efficiencies in excess of 81% for all the higher-order modes are obtained in the wavelength range 1530-1600 nm. For the LP01, LP11, LP21, LP02, LP31, LP12 MSCs at 1550 nm, the coupling ratios are 96.2%, 99.8%, 89.5%, 85.0%, 90.9%, 96.1%, respectively, and the mode purity of the MSCs is higher than 88.0%. The loss of the MSCs is lower than 1.8 dB in the wavelength range 1530-1600 nm. This device can be applied in broadband mode-division multiplexing transmission systems.