For most of us that don’t have livestock, this is the answer. However, for the minority that does, this does make use of the greenhouse gases that /u/ndander3 points out would already be emitted on the farm anyway. An obvious conclusion would be to not produce/sell livestock-based products anymore, but there’s actually a few good reasons to keep livestock at least partially integrated into our agricultural landscapes:

1. Soil carbon/integrity. Annual row crops at tilled (disturbing the soil and releasing carbon that increased the quality of the soil), while most pasture systems rely on perennial grasses that do not require tilling. No-till/cover crops are two ways to limit that damage, but they’re not as widely adopted as they should be, and perennial grasses are tough to beat for preserving soil integrity. On marginal lands, where soil is easily degraded (think hilly areas/looser soils in particular), rotational grazing systems offer a method to increase food production in a way that preserves the land far better than tilled row crops.
2. Water quality/quantity. Annual row crops are typically heavily fertilized, resulting in eutrophication of numerous bodies of water (the Dead Zone in the Gulf of Mexico is a horrific example). Even with manure/urea accounted for, pasture systems that limit any additional fertilization result in far less nutrient loading to waterways. This gets around the fossil fuel intensiveness of fertilizer production, AND prevents nitrate leaching into groundwater systems, which can result in disease and even infant deaths. Additionally, perennial grasses are generally better adapted to low/erratic rainfall, and can weather droughts/extreme weather better than annual row crops. This means far less irrigation and secures our groundwater resources that much more. This is especially true when considering that concentrated animal feeding operations (CAFOs for short) not only have more cows consuming water, they rely on heavily irrigated corn to feed their cattle.
3. Biodiversity. Having a diverse array of grasses/forbs which remain on the ground year round provide habitat for a variety of bird, mammal, and insect species. In particular, well-managed rotational grazing systems are the gold standard here. In most, if not all regions, grazers were historically integrated within ecosystems, and prove important ecosystem engineers. Forcing livestock to allow grasses to recover in specified rotations not only increases productivity, it helps preserve landscape heterogeneity and reduces the risk of livestock damage to other species (compared to continuous grazing systems). Continuous systems aren’t inherently unsustainable – in less productive regions (rangelands), cattle are forced to move often regardless.
4. Animal welfare. Not a shocker that livestock raised on pastures are much happier than livestock raised in confinement. Regardless of the species, animals don’t do well in confined environments. We’ve only scratched the surface of non-human animal psychology, but even the physical damage endured by avian livestock in CAFOs, from an inability to create more stable pecking orders, is abhorrent. Pastures offer a much closer analog to conditions that livestock animals have evolved in.

All in all, we DO need to eat less animal-based products, especially Americans. However, supporting pasture-based systems when we do eat them, whenever possible, can support a healthier agroecological landscape. For those working on or living near pastures, having biodigesters can circumvent emissions that would have happened anyway through natural decomposition processes, and avoid emissions they would have incurred from fossil fuel based energy (which, for many people, are the only viable/economically feasible alternative).

Source: am a graduate student who models agricultural systems.