Dear Elena:
The phrase "pleading the blood" was perhaps popularized by Pentecostals and some fundamentalists, but it first has a tradition in Catholic prayer. Like many things that non-Catholic think they invented, it was first invented by Catholics long before Protestants existed.
The blood of Jesus represents the sacrifice of Christ on the Cross shedding his blood to redeem mankind from the realm of Satan and to wash away our sins. The blood in the Old Testament was also an instrument of cleansing, and a mark of the mercy and protection of God (such as on Passover night). As such, demons are particularly frightened of the blood of Jesus, hence the reference to His blood in prayer.
One such prayer is the Litany of the Precious Blood of Jesus:
Lord, have mercy.
Christ, have mercy.
Lord, have mercy. Christ, hear us.
Christ, graciously hear us.
God the Father of Heaven,
have mercy on us.
God the Son, Redeemer of the world,
have mercy on us.
God, the Holy Spirit,
have mercy on us.
Holy Trinity, One God,
have mercy on us.
R. Save Us
Blood of Christ, Incarnate Word of God...
Blood of Christ, of the New and Eternal Testament...
Blood of Christ, falling upon the earth in the Agony...
Blood of Christ, shed profusely in the Scourging...
Blood of Christ, flowing forth in the Crowning with Thorns...
Blood of Christ, poured out on the Cross...
Blood of Christ, Price of our Salvation...
Blood of Christ, without which there is no salvation...
Blood of Christ, Eucharistic drink and refreshment of souls...
Blood of Christ, river of Mercy...
Blood of Christ, Victor over demons...
Blood of Christ, Courage of Martyrs...
Blood of Christ, Strength of Confessors...
Blood of Christ, bringing forth Virgins...
Blood of Christ, Help of those in peril...
Blood of Christ, Relief of the burdened...
Blood of Christ, Solace in sorrow...
Blood of Christ, Hope of the penitent...
Blood of Christ, Consolation of the dying...
Blood of Christ, Peace and Tenderness of hearts...
Blood of Christ, Pledge of Eternal Life...
Blood of Christ, freeing souls from Purgatory...
Blood of Christ, most worthy of all glory and honor...
Lamb of God, Who takes away the sins of the world,
spare us, O Lord.
Lamb of God, Who takes away the sins of the world,
graciously hear us, O Lord
Lamb of God, Who takes away the sins of the world,
have mercy on us.
Thou has redeemed us, O Lord, in Thy Blood,
And made of us a kingdom for our God.
Let Us Pray
Almighty and Eternal God, Thou hast appointed Thine Only-Begotten Son the Redeemer of the world, and willed to be appeased by His Blood. Grant, we beseech Thee, that we may worthily adore this Price of our Salvation, and through Its Power be safeguarded from the evils of this present life, so that we may rejoice in Its Fruits forever in Heaven. Through the same Christ our Lord. Amen.
To "plead the blood" on one's house or household would be the equivalent of the blood on the doors on Passover Night which saved the first born of those inside the house from the Angel of Death — a symbol to the evil forces that this house belongs to God. As Catholics do much the same thing by blessing a house with prayers and holy water and blessed medals and such.
I use phrases in spiritual warfare prayers such as "by the shed Blood of Jesus Christ" and "by the virtue of the shed Blood of our Lord Jesus" and "purchased by the blood of Jesus Christ" and "I am now cleansed by the blood of Jesus Christ" and "protection of the Blood of Jesus Christ and in at least one place I use the term "plead the blood of Jesus Christ over our children". We ought to "plead the blood", that is, to ask for the protection of the Blood of Jesus Christ, over our families.
The Blood (of Christ) is a cleansing and a protection. The Bible has many examples of the importance of the Blood (e.g., Exodus 12:13; Leviticus 17:11; Matthew 26:28; Ephesians 1:7; 2:13; Colossians 1:13; and Revelation 12:11).
So the answer is that to "plead the blood", or words to that effect, does have its place, has its basis in Biblical principles, and has similar prayers in traditional Catholic prayers, such as the Litany of the Precious Blood. To "plead", by the way simply means "to appeal or entreat earnestly."
Whoever said that "pleading the blood" is blasphemous is quiet ignorant. Ignore them.
God Bless,
Bro. Ignatius Mary