Hose down the patio and the furniture. Spider mites exceptionally tiny and cannot withstand the water pressure coming from the hose.
Remove any potted plants you have on the patio table, the source of food for the spider mites. Check the pots and plants for infestations and quarantine until the mites disappear or you have eradicated the spiders.
Spray your patio furniture with insecticidal oils or soaps. Many bug sprays do not kill spider mites, so when choosing a horticultural oil or soap, check the label to verify that the product works on red spider mites. The sprays and soaps do not have any residual toxicity, so you have to reapply if you see the mites on the furniture later.
Wrap the base of the chairs and table with double-sided sticky tape. The spider mites climb up from the ground a tacky barrier keeps them from progressing up your furniture. While spider mites lay their eggs on plants, your patio furniture is not a place they are likely to deposit their eggs. Unless you brought the mites home on a plant for the patio table, the only way that the spider mites reach your furniture is by climbing up from the surface of the patio.