Throat and nasal membranes damaged by dry air become more vulnerable to cold viruses. It's essential to use heaters for cold prevention and humidifiers for dryness prevention.

Literal

Dry air [because-のせいで] damaged throat [and-や] nose [possessive-の] mucous-membranes [topic-は], cold [possessive-の] virus [subject-が] enter-[easy-やすく] become-[regret-てしまいます]. Heater [with-で] cold-countermeasure, humidifier [with-で] dryness-countermeasure [object-を] firmly carry-out [nominalizer-こと] [subject-が] essential [copula-です].

A health-advice passage using parallel structure: 暖房器具で寒さ対策、加湿器で乾燥対策 mirrors X で Y 対策 twice. 肝要 ('essential, crucial') is a formal kango word — 肝 'liver' + 要 'vital' — reflecting classical Chinese medical tradition. 入り込みやすくなってしまいます chains three auxiliaries: easy-to-enter + become + unfortunately.