The Covid-19 pandemic has abruptly changed well-established mobility patterns, as the need for social distancing and lockdown orders have driven citizens to reduce their movements and avoid crowded mass transit. In this context, we look at the case of New York City's bike sharing system, one of the largest in the world, to gain insights on the socioeconomic variables behind urban mobility during a pandemic. We exploit several sources of Smart City data to analyze the relationship between bike sharing, public transport, and other modes of transportation, deriving interesting insights for future urban planning, both city-wide and at the neighborhood level. The New York City case study shows some of the most important trends during the lockdown, and the combination between mobility and socioeconomic data can be used to understand the consequences of the pandemic on different communities, as well as the future directions of expansion and management of the bike sharing system and urban infrastructure.