Planets that form around stars born in dense stellar environments are subject to dynamical perturbations from other stars in the system. These perturbations will strip outer planets, forming a population of free-floating planets, some of which will be tidally captured before they evaporate from the system. For systems with velocity dispersion of 1 km/s, Jupiter-mass planets can be captured into orbits with periods of 0.1-0.4 days, which are generally stable over a Gyr, assuming quadratic suppression of eddy viscosity in the convective zones of the host stars. Under this assumption, and that most stars form several massive planets at separations 5-50 AU, about 0.03% of stars in rich, mature open clusters should have extremely close-in tidally captured planets. Approximately 0.005% of field stars should also have such planets, which may be found in field searches for transiting planets. Detection of a population of tidally-captured planets would indicate that most stars formed in stellar clusters. In globular clusters, the fraction of stars with tidally-captured planets rises to 0.1% -- in conflict with the null result of the transit search in 47 Tuc. This implies that, if the quadratic prescription for viscosity suppression is correct, planetary formation was inhibited in 47 Tuc: less than one planet of Jupiter-mass or greater (bound or free-floating) formed per cluster star. Less than half of the stars formed solar-system analogs. Brown dwarfs can also be captured in tight orbits; the lack of such companions in 47 Tuc in turn implies an upper limit on the initial frequency of brown dwarfs in this cluster. However, this upper limit is extremely sensitive to the highly uncertain timescale for orbital decay, and thus it is difficult to draw robust conclusions about the low-mass end of the mass function in 47 Tuc.