Saptarshis live for a period of Manvantara. So, they are alive now. They live on seven stars (Sapta Rishi Mandala) which revolve around Pole Star.
Srimad Bhagavatam 8.13.5 gives names of seven sages for Vaivasvara Manvantara.
कश्यपोऽत्रिर्वसिष्ठश्च विश्वामित्रोऽथ गौतमः|
सप्तर्षयः स्मृताः ॥ ८.१३.५ ॥
Kaśyapa, Atri, Vasiṣṭha, Viśvāmitra, Gautama, Jamadagni and Bharadvāja
are known as the seven sages.
It also mentions names of Sapta rishis of Savarni Manvantara which is next Manvantara.
गालवो दीप्तिमान्रामो द्रोणपुत्रः कृपस्तथा|
भगवान्बादरायणः ॥ ८.१३.१५ ॥
इमे सप्तर्षयस्तत्र भविष्यन्ति स्वयोगतः|
इदानीमासते राजन्स्वे स्व
आश्रममण्डले ॥ ८.१३.१६ ॥
O King, during the eighth manvantara, the great personalities Gālava,
Dīptimān, Paraśurāma, Aśvatthāmā, Kṛpācārya, Ṛṣyaśṛṅga and our father,
Vyāsadeva, the incarnation of Nārāyaṇa, will be the seven sages. For
the present, they are all residing in their respective āśramas.
During onset of every Satya Yuga, Sapta Rishis come to earth and recount Dharma that is lost (by end of Kali Yuga all Dharma [hindu principles] will be lost). This is mentioned in Chapter 31 Anusanga of Brahmanda Purana.
103-106. Owing to the power of the inevitable future, the Krta Yuga
set in. When the holy Krta Yuga began to function, the subjects
pertaining to the Krta Yuga were born of those who survived from the
Kali Yuga. Those Siddhas who remained then, began to move about
invisibly. They established themselves there along with the seven
sages. The Brähmanas, the Ksatriyas, the Vaisyas and the Südras
were those, who were remembered as seeds. (They were intended to be
the nucleus for the succeeding generation). At that time they became
identical with those who were born in ( the closing stages of) Kali.
They were devoid of any difference. The seven sages recounted
Dharma to them as well as to others.
107.That ( Dharma) is equipped with the conduct of life of the different varnas and stages. It is of two types : Srauta (as laid
down in the Vedas) and the Smarta (as laid down in the Smrtis). The
object in the Krta age remained thus as remained till the close of the
Yuga for the sake of establishing Dharma.
109-110. The sages remain in authority over the whole of the