One of the main difficulties in using older stromatolitic deposits, asides from the sparcity of deposits , is the debate of their biogenesis, particularly those from the Archaean (1). However, 3.45 Ga coniform stromatolite deposits from the Pilbara region of Western Australia (with similar features as those already identified as biogenic), suggest that they were formed in a hypersaline depositional environment (2). Warm, even arid environments are suggested by authors such as Chumakov (3) and Medvedev et al. (4). Below are a couple of examples alongside the paper that plannapus provided in the comment to the other answer.
One of the best preserved stromatolitic suites is, however, from the the Late Archaean (2.7 Ga) stromatolitic limestones within sulphide rich and tidal-flat sediments of the Manjeri and Cheshire Formations in the Belingwe Greenstone Belt of Zimbabwe (5), (6). The authors observe that the stromatolites, in conjuction with the sulphides and tidal flat deposits indicate that there was a thriving prokaryotic community of oygenetic photosynthesisers and sulphate reducers, also with possible methanogenesis and methanotrophy - the implication according to the authors is that these suggest well developed carbon and sulfur cycles - that could have been global.
Microstromatolitic deposits from the 2.2-2.1 Ga carbon excursion described by Medvedev et al. 2005 (4) have been found alongside other inclusions that indicate that the environment of their formation was warm and arid.
Within the research, it becomes clear that stromatolitic deposits, alongside other sedimentary structures and included isotopes are a reasonable way to gauge the climatic and even, at least part, of the biogeochemical cycles of the time. However, as in the Memoirs edited by Kesler and Ohmoto (6), greater constraints of the conditions can be made using the mineralogical ore deposits within the ancient cratons.
Note: Ga = billion years
(1) Batchelor et al. 2004 A case for biotic morphogenesis of coniform
(2) Hofmann et al. 1999 Origin of 3.45 Ga coniform stromatolites in Warrawoona Group, Western Australia, Geological Society of America Bulletin
(3) Chumakov 2003 Global climates of the Vendian Russian Journal of Earth Sciences
(4) Medvedev et al. 2005 Testing the Biostratigraphic Potential of Early Palaeoproterozoic Microdigitate Stromatolites Revista Española de Micropaleontología
(5) Grassineau et al. 2002 Stable isotopes in the Archaean Belingwe belt, Zimbabwe: evidence for a diverse microbial mat ecology The Geological Society of London
(6) Kesler and Ohmoto, 2007 Evolution of Early Earth's Atmosphere, Hydrosphere, and Biosphere - Constraints from Ore Deposits Memoir of the Geological Society of America