Nanocomposites based on poly(styrene-block-isobutylene-block-styrene) (SIBS) and single-walled carbon nanotubes (CNTs) were prepared and characterized in terms of tensile strength as well as bio- and hemocompatibility. It was shown that modification of CNTs using dodecylamine (DDA), featured by a long non-polar alkane chain, provided much better dispersion of nanotubes in SIBS as compared to unmodified CNTs. As a result of such modification, the tensile strength of the nanocomposite based on SIBS with low molecular weight (Mn = 40,000 g mol–1) containing 4% of functionalized CNTs was increased up to 5.51 ± 0.50 MPa in comparison with composites with unmodified CNTs (3.81 ± 0.11 MPa). However, the addition of CNTs had no significant effect on SIBS with high molecular weight (Mn~70,000 g mol−1) with ultimate tensile stress of pure polymer of 11.62 MPa and 14.45 MPa in case of its modification with 1 wt% of CNT-DDA. Enhanced biocompatibility of nanocomposites as compared to neat SIBS has been demonstrated in experiment with EA.hy 926 cells. However, the platelet aggregation observed at high CNT concentrations can cause thrombosis. Therefore, SIBS with higher molecular weight (Mn~70,000 g mol−1) reinforced by 1–2 wt% of CNTs is the most promising material for the development of cardiovascular implants such as heart valve prostheses.