We study the interplanetary magnetic field (IMF) data obtained by the Cassini spacecraft during a ~6.5-month interval when the spacecraft was approaching Saturn at heliocentric distances between ~8.5 and ~8.9 AU. It is shown that the structure of the IMF is consistent with that expected to be formed by corotating interaction regions (CIRs) during the declining phase of the solar cycle, with two sectors during each solar rotation, and crossings of the heliospheric current sheet generally embedded within few-day higher-field compression regions, separated by several-day lower-field rarefaction regions. This pattern was disrupted in November 2003, however, by an interval of high activity on the Sun. These data have then been employed to estimate the voltage associated with open flux production at Saturn's magnetopause using an empirical formula adapted from Earth. The results show that the CIR-related structuring of the IMF leads to corresponding structuring of the interplanetary interaction with Saturn's magnetosphere and hence also to intervals of very different dynamical behavior. During few-day compression regions where the IMF strength is ~0.5--2 nT, the average Dungey cycle voltage is estimated to be ~100 kV, such that the open flux produced over such intervals is ~30--40 GWb, similar to the typical total amount present in Saturn's magnetosphere. The magnetosphere is thus significantly driven by the solar wind interaction during such intervals. During some rarefaction intervals, on the other hand, the field strength remains ~0.1 nT or less over several days, implying reconnection voltages of ~10 kV or less, with negligible production of open flux. The magnetosphere is then expected to enter a quiescent state, dominated by internal processes. Overall, ~100 GWb of open flux is estimated to be produced during each ~25-day solar rotation, about 3 times the typical flux contained in the tail, and sufficient to drive three to five substorms. We point out, however, that CIR-related variations in solar wind dynamic pressure will also occur in synchronism with the field variations, which may also play a role in modulating the open flux in the system, thus reinforcing the synchronization of the pattern of growth and decay of open flux to the CIR pattern. Estimates of open flux production associated with the period of strong solar activity indicate that major magnetospheric dynamics were excited by reconnection-mediated solar wind interaction during this interval.