We have made new and improved measurements of the neutron capture and total cross sections for 88Sr at the Oak Ridge Electron Linear Accelerator (ORELA). Improvements over previous measurements include a wider incident neutron energy range, better resolution, the use of metallic rather than carbonate samples, better background subtraction, reduced sensitivity to sample-dependent backgrounds, and better pulse-height weight-ing functions. Because of its small cross section, the 88Sr(n,γ) reaction is an important bottleneck during s-process nucleosynthesis. Hence, an accurate determination of this rate is needed to better constrain the neutron exposure in s-process models and to better understand the recently discovered isotopic anomalies in certain meteorites. We performed an R-matrix analysis of our capture and transmission data to extract parameters for 101 resonances between 100 eV and 350 keV. In addition, we fitted our transmission data alone to extract parameters for 342 additional resonances between 350 and 950 keV. We used this information to calculate average properties of the 88Sr+n system for comparison to previous work. Although previous data and resonance analyses were much less extensive, they are, in general, in good agreement with our results except that the average radiation widths as well as the p-wave correlation coefficients we determined are significantly smaller, and the s-wave correlation coefficient we determined has opposite sign from that reported in previous work. We used these resonance parameters together with a calculation of the small, but significant direct-capture contribution to determine the astrophysical reaction rate for the 88Sr(n,γ) reaction to approximately 3% accuracy across the entire range of temperatures needed by s-process models. Our new rate is in good agreement with the results from a high-precision activation measurement at kT=25 keV, but it is approximately 9.5% lower than the rate used in most previous nucleosynthesis calculations in the temperature range (kT=6-8 keV), where most of the neutron exposure occurs in current stellar models of the s process. We discuss the possible astrophysical impact of this new, lower rate.